Zero: The Biography of a Dangerous Idea

by Charles Seife

Take any term and divide it by its previous term. For instance, 8/5=1.6; 13/8=1.625; 21/13=1.61538 . . . . These ratios approach a particularly interesting number: the golden ratio, which is 1.61803 . . . .

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.*

The Arabic notation was allowed into Italy and soon spread throughout Europe. Zero had arrived—as had the void.

Before the fifteenth century, paintings and drawings were largely flat and lifeless. The images in them were distorted and two-dimensional;

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.)

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

The vanishing point turned a two-dimensional drawing into a perfect simulation of a three-dimensional building.

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.

singularity, a concept that became very important later in the history of science—but

“Let no one who is not a mathematician read my works.”

Zero had transformed the art world.

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 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.

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.

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.

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.

Zero and the infinite were at the very center of the philosophical war taking place during the sixteenth and seventeenth centuries.

Like Pythagoras, Descartes was a mathematician-philosopher; perhaps his most lasting legacy was a mathematical invention—what we now call Cartesian coordinates.

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.)

Descartes quickly realized how powerful his coordinate system was. He used it to turn figures and shapes into equations and numbers; with Cartesian coordinates every geometric object—squares, triangles, wavy lines—could be represented by an equation, a mathematical relationship.

To Descartes, zero was also implicit in God’s domain, as was the infinite. Since the old Aristotelian doctrine was crumbling, Descartes, true to his Jesuit training, tried to use nought and infinity to replace the old proof of God’s existence.

Learning is just the process of uncovering that previously imprinted code of laws about the workings of the universe.

Even today, children are taught “Nature abhors a vacuum,” while the teachers don’t really understand where that phrase came from. It was an extension of the Aristotelian philosophy: vacuums don’t exist. If someone would attempt to create a vacuum, nature would do anything in its power to prevent it from happening. It was Galileo’s secretary, Evangelista Torricelli, who proved that this wasn’t true—by creating the first vacuum.

In 1643, Torricelli took a long tube that was closed at one end and filled it with mercury. He upended it, placing the open end in a dish also filled with mercury. Now, if Torricelli had upended the tube in air, everyone would expect the mercury to run out, because it would quickly be replaced by air; no vacuum would be created. But when it was upended in a dish of mercury, there was no air to replace the mercury in the tube. If nature truly abhorred a vacuum so much, the mercury in the tube would have to stay put so as not to create a void. The mercury didn’t stay put. It sank downward a bit, leaving a space at the top. What was in that space? Nothing. It was the first time in history anyone had created a sustained vacuum.

In 1623, Descartes was twenty-seven, and Blaise Pascal, who would become Descartes’s opponent, was zero years old. Pascal’s father, Étienne, was an accomplished scientist and mathematician; the young Blaise was a genius equal to his father. As a young man, Blaise invented a mechanical calculating machine, named the Pascaline, which is similar to some of the mechanical calculators that engineers used before the invention of the electronic calculator.

About the time of the Pascals’ conversion, a friend of Étienne’s—a military engineer—came to visit and repeated Torricelli’s experiment for the Pascals. Blaise Pascal was enthralled, and started performing experiments of his own, using water, wine, and other fluids. The result was New experiments concerning the vacuum, published in 1647. This work left the main question unanswered: why would mercury rise only 30 inches and water only 33 feet? The theories of the time tried to save a fragment of Aristotle’s philosophy by declaring that nature’s horror of the vacuum was “limited”; it could only destroy a finite amount of vacuum. Pascal had a different idea.

To Pascal, this seemingly bizarre behavior proved that it wasn’t an abhorrence of the vacuum that drove the mercury up the tube. It was the weight of the atmosphere pressing down on the mercury exposed in the pan that makes the fluid shoot up the column. The atmospheric pressure bearing down on a pan of liquid—be it mercury, water, or wine—will make the level inside the tube rise, just as gently squeezing the bottom of a toothpaste tube will make the contents squirt out the top. Since the atmosphere cannot push infinitely hard, it can only drive mercury about 30 inches up the tube—and at the top of the mountain, there is less atmosphere pushing down, so the air can’t even push the mercury as high as 30 inches.

nature has no repugnance for the vacuum, that it makes no effort to avoid it, and that it admits vacuum without difficulty and without resistance.” Aristotle was defeated, and scientists stopped fearing the void and began to study it.

Deep within the scientific world’s powerful new tool—calculus—was a paradox. The inventors of calculus, Isaac Newton and Gottfried Wilhelm Leibniz, created the most powerful mathematical method ever by dividing by zero and adding an infinite number of zeros together. Both acts were as illogical as adding 1 + 1 to get 3. Calculus, at its core, defied the logic of mathematics. Accepting it was a leap of faith. Scientists took that leap, for calculus is the language of nature. To understand that language completely, science had to conquer the infinite zeros.

harmonic series: 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + . . .

Like the Zeno sequence and Suiseth’s sequence, all the terms get closer and closer to zero. However, when Oresme tried to sum the terms in the sequence, he realized that the sums got larger and larger and larger. Even though the individual terms go to zero, the sum goes off to infinity. Oresme showed this by clumping the terms together: 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + . . . . The first group clearly equals 1/2; the second group is greater than (1/4 + 1/4), or 1/2. The third group is greater than (1/8 + 1/8 + 1/8 + 1/8), or 1/2. And so forth. You keep adding 1/2 after 1/2 after 1/2, and the sum gets bigger and bigger, and off to infinity. Even though the terms themselves go to zero, they don’t approach zero fast enough. An infinite sum of numbers can be infinite, even if the numbers themselves approach zero. Yet this isn’t the strangest aspect of infinite sums. Zero itself is not immune to the bizarre nature of infinity.

An Italian priest, Father Guido Grandi, even used this series to prove that God could create the universe (1) out of nothing (0).

Sometimes, when the terms go to zero, the sum is finite, a nice, normal number like 2 or 53. Other times the sum goes off to infinity. And an infinite sum of zeros can equal anything at all—and everything at the same time.

Kepler, at least, wasn’t afraid of a glaring problem: as Δx goes to zero, the sum becomes equivalent to adding an infinite number of zeros together—a result that makes no sense. Kepler ignored the problem; though adding infinite zeros together was gibberish from a logical point of view, the answer it yielded was the right one.

To Cavalieri, every area, like that of the triangle, is made up of an infinite number of zero-width lines, and a volume is made up of an infinite number of zero-height planes. These indivisible lines and planes are like atoms of area and volume; they can’t be divided any further. Just as Kepler measured the volumes of barrels with his thin slices, Cavalieri added up an infinite number of these indivisible zeros to figure out the area or the volume of a geometric object.

infinite zeros make no logical sense.

This means that the difference in height, Δy, goes to zero, as does the horizontal distance between the points, Δx. As your tangent approximations get better and better, Δy/Δx approaches 0/0. Zero divided by zero can equal any number in the universe. Does the slope of the tangent line have any meaning?

If, as Newton insisted, (oẋ)2 and (oẋ)3 and higher powers of oẋ were equal to zero, then oẋ itself must be equal to zero.* On the other hand, if oẋ was zero, then dividing by oẋ as we do toward the end is the same thing as dividing by zero—as is the very last step of getting rid of the o in the top and bottom of the oẏ/oẋ expression. Division by zero is forbidden by the logic of mathematics.

Calculus is the combination of these two tools, differentiation and integration, in one package. Though Newton broke some very important mathematical rules by toying with the powers of zero and infinity, calculus was so powerful that no mathematician could reject it.

These early equation-laws were extremely good at expressing simple relationships, but equations have limitations—their constancy, which prevented them from being universal laws.

Calculus allowed Newton to combine all these equations into one grand set of laws—laws that applied in all cases, under all conditions. For the first time, science could see the universal laws that underlie all of these little half laws. Even though mathematicians knew that calculus was deeply flawed—thanks to the mathematics of zero and infinity—they quickly embraced the new mathematical tools. For the truth is, nature doesn’t speak in ordinary equations. It speaks in differential equations, and calculus is the tool that you need to pose and solve these differential equations.

Differential equations are not like the everyday equations that we are all familiar with. An everyday equation is like a machine; you feed numbers into the machine and out pops another number. A differential equation is also like a machine, but this time you feed equations into the machine and out pop new equations.

It is a universal law. It is a glimpse at the machinery of nature.

simple differential equation that describes the motion of all objects in the universe: F=mẋ, where F is the force on an object and m is its mass.

If you’ve got an equation that tells you about the force that is being applied on an object, the differential equation reveals exactly how the object moves.

Looking back, it appears that Leibniz formulated his version independently of Newton, though the matter is still being debated. The two had a correspondence in the 1670s, making it very difficult to establish how they influenced each other. However, though the two theories came up with the same answers, their notations—and their philosophies—were very different.

In a sense these infinitesimals were infinitely small, smaller than any positive number you could name, yet still somehow greater than zero.

underneath all the mathematics, Leibniz’s differentials still had the same forbidden 0/0 nature that plagued Newton’s fluxions. As long as this flaw remained, calculus would be based upon faith rather than logic. (In fact, faith was very much on Leibniz’s mind when he derived new mathematics, such as the binary numbers. Any number can be written as a string of zeros and ones; to Leibniz, this was the creation ex nihilo, the creation of the universe out of nothing more than God/1 and void/0. Leibniz even tried to get the Jesuits to use this knowledge to convert the Chinese to Christianity.)

Guillaume-François-Antoine de l’Hôpital