Zero: The Biography of a Dangerous Idea

by Charles Seife

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.

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,

The church had other tools to fight heresy as well; the Spanish Inquisition started burning Protestants in 1543,

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.

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.

astrologermonk, 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.

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.

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.

No longer were the Western art of geometry and the Eastern art of algebra separate domains.

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

Pascal was a mathematician as well as a scientist. In science Pascal investigated the vacuum—the nature of the void. In mathematics Pascal helped invent a whole new branch of the field: probability theory. When Pascal combined probability theory with zero and with infinity, he found God.

Pascal argued that it was best to believe in God, because it was a good bet. Literally.

Just as he analyzed the value—or expectation—of a gamble, Pascal analyzed the value of accepting Christ as savior. Thanks to the mathematics of zero and infinity, Pascal concluded that one should assume that God exists.

Nobody was willing to say that there was zero chance that God exists. No matter what your outlook, it is always better to believe in God, thanks to the magic of zero and infinity. Certainly Pascal knew which way to wager, even though he gave up mathematics to win his bet.

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.

Johannes Kepler—the man who figured out that planets move in ellipses—spent that year gazing into wine barrels, since he realized that the methods that vintners and coopers used to estimate the size of barrels were extremely crude.

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.

For geometers, Cavalieri’s statement was troublesome indeed; adding infinite zero-area lines could not yield a two-dimensional triangle, nor could infinite zero-volume planes add up to a three-dimensional structure. It was the same problem: infinite zeros make no logical sense.

For this reason, several seventeenth-century mathematicians—like Evangelista Torricelli, René Descartes, the Frenchman Pierre de Fermat (famous for his last theorem),

and the Englishman Isaac Barrow—created different methods for calculating the tangent line to any given point on a curve. However, like Cavalieri, all of them came up against the infinitesimal.

As your tangent approximations get better and better, Δy/Δx approaches 0/0. Zero divided by zero can equal any number in the universe.

The first discoverer of calculus nearly died before he ever took a breath. Born prematurely on Christmas Day in 1642, Isaac Newton squirmed into the world, so small that he was able to fit into a quart pot. His father, a farmer, had died two months earlier.

This process, the first half of calculus, is now known as differentiation; however, Newton’s method of differentiation doesn’t look very much like the one we use today.

The method gave the right answer, but Newton’s vanishing act was very troubling. 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.*

Newton’s method of fluxions was very dubious. It relied upon an illegal mathematical operation, but it had one huge advantage. It worked. The method of fluxions not only solved the tangent problem, it also solved the area problem. Finding the area under a curve (or a line, which is a type of curve)—an operation we now call integration—is

The process of differentiation destroys information, so the process of integration doesn’t give you exactly the answer you are looking for unless you add another bit of information.) Calculus is the combination of these two tools, differentiation and integration,

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.

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...

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 machin...

equation is also like a machine, but this time you feed equations into the machine a...