Zero: The Biography of a Dangerous Idea

“The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, of religion. Underneath every revolution lay a zero—and an infinity.”

Seife emphasizes the importance of zero across all human disciplines and areas of inquiry. He also emphasizes that zero was a matter of contention that constantly destabilized human thought.

“Yet through all its history, despite the rejection and the exile, zero has always defeated those who opposed it. Humanity could never force zero to fit its philosophies. Instead, zero shaped humanity’s view of the universe—and of God.”

Seife depicts zero as an agent in its own right and suggests that it is like a supernatural messenger or holy book imparting transcendental truths, introducing the idea of The Revelation of Zero. Any human thought might be matched by another human thought, and many human ideas can be adjusted to fit into new philosophies, but zero’s invincibility and uncompromising strangeness make it special, Seife suggests.

“It’s difficult for a modern person to imagine a life without zero, just as it’s hard to imagine life without the number seven or the number 31. However, there was a time where there was no zero, just as there was no seven and 31.”

Modern readers will likely take zero for granted. Seife encourages his readers to engage their imagination because he wants them to experience wonder. Zero is strange, like a mythological creature.

“Multiplying by zero collapses the number line. But dividing by zero destroys the entire framework of mathematics. There is a lot of power in this simple number. It was to become the most important tool in mathematics. But thanks to the odd mathematical and philosophical properties of zero, it would clash with the fundamental philosophy of the West.”

Seife emphasizes the destructive power of zero—i.e., The Peril of Zero. He sets zero in opposition to Western thought with the choice word “clash”; the history of zero is a history of struggle in which zero ultimately prevails, as Seife foreshadows.

“Zero had no place within the Pythagorean framework. The equivalence of numbers and shapes made the ancient Greeks the masters of geometry, yet it had a serious drawback. It precluded anyone from treating zero as a number. What shape, after all, could zero be?”

Seife gives a conceptual explanation for why the Greeks, and by extension Western thought, rejected zero: Shapes with zero area and other similar geometric implications are counterintuitive or outright absurd. Seife emphasizes that the Greeks knew about zero and consciously rejected it out of fear.

“Aristotle hated the idea of the void so much that he chose the eternal, infinite universe over one that had a vacuum in it […] The Aristotelian view of physics, as wrong as it was, was so influential that for more than a millennium it eclipsed all opposing views […] Science would never progress until the world discarded Aristotle’s physics.”

The long reign of Aristotelianism and its bias against zero is a dominant aspect of Seife’s account of Eastern and Western thought’s conflicting perspectives on zero. However, Seife gradually narrows his focus to Western thought alone, and then the conflict becomes an internal one. Aristotelianism remains the primary antagonist of zero throughout the entire book.

“Medieval scholars branded void as evil—and evil as void. Satan was quite literally nothing. Boethius made the argument as follows: God is omnipotent. There is nothing God cannot do. But God, the ultimate goodness, cannot do evil. Therefore evil is nothing. It made perfect sense to the medieval mind.”

Seife demonstrates the bias of Western thought against zero. Zero was considered not merely difficult but unsafe, frightening, and corrupt, to the extent of being associated with the devil himself. In his exploration of the beliefs and emotions surrounding zero, Seife aims to engage readers more than a mere sequence of mathematical discoveries might.

“[Nishkala Shiva] was the ultimate void, the supreme nothing—lifelessness incarnate. But out of the void, the universe was born, as was the infinite. Unlike the Western universe, the Hindu cosmos was infinite in extent; beyond our own universe were innumerable other universes. At the same time, though, the cosmos never truly abandoned its original emptiness.”

In describing this central Hindu god, Seife emphasizes the dualism of zero and infinity and how Hinduism’s acceptance of zero gave it a more accurate view of the universe. The Hindu ideas Seife describes have seemingly prescient similarities to later scientific discoveries, including the Big Bang and the heat death of the universe. This is the revelation of zero: It allowed people to conceptualize certain scientific concepts far in advance of modern science.

“[This] was zero. Not just a mere placeholder zero that represents an empty space on the abacus, but zero the number. It had a specific value, a fixed place on the number line. […] Zero had finally arrived. However, even the Indians thought that zero was a pretty bizarre number, for all the usual reasons.”

Seife clarifies that even in receptive cultures zero remained untamed, a supernatural power that held promise and peril. That Seife considers India the birthplace of the mathematical zero only makes this point more emphatic.

“Maimonides argued that there were flaws [with Aristotle’s eternal universe]. After all, it conflicted with the Scriptures. […] Aristotle had to go. Maimonides stated that the act of creation came from nothing. It was creatio ex nihilo, despite the Aristotelian ban on the vacuum. With that stroke the void moved from sacrilege to holiness.”

“Scripture” traditionally refers to divine revelation; Seife mentions Maimonides’s argument to give zero a touch of transcendence that will linger, giving Seife’s discussion of the Big Bang a mixed flavor of science and mysticism related to the idea of zero as revelatory.

“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.”

Seife associates the acceptance of zero with intellectual flourishing in the Renaissance and the Scientific Revolution, further underscoring zero’s ability to unlock the universe’s truths. Seife specifies that Aristotelianism, not mere religion or even Christianity, was the enemy of zero. Descartes and Pascal, two key figures in this chapter, at last integrated zero and infinity into their theological reasonings, with (Seife implies) positive results.

“Descartes unified numbers and shapes. No longer were the Western art of geometry and the Eastern art of algebra separate domains. […] Zero was at the center of the coordinate system, and zero was implicit in each geometric shape.”

Seife emphasizes just how crucial to zero’s acceptance in the West Descartes’s coordinate plane was. It overcame the apparent divide between shapes and numbers that biased Pythagorean reasoning against zero, and it visualized zero’s central location on the number line in an intuitive way.

“Pascal argued that it was best to believe in God, because it was a good bet. Literally. Just as he analyzed the value—of 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.”

Seife lingers on Pascal’s probabilistic argument for believing in God because it is a clear instance of zero having significance as a source of insight—a gateway to transcendental truths. Seife will later substitute “the magic of zero and infinity” for “the mathematics of zero and infinity” (104), heightening the sense of zero’s supernatural or even divine powers.

“Every time mathematicians tried to deal with the infinite or with zero, they encountered trouble with illogic. To figure out the volume of a barrel or the area under a parabola, mathematicians added infinite zeros together; to find out the tangent of a curve, they divided zero by itself.”

Seife gives a concise explanation of why precisely zero bothered mathematicians: It concealed the solutions to important problems behind absurd operations. This is related to the peril of zero.

“Nobody worried about dividing by zero when conveniently ignoring the rules of mathematics explained everything from the fall of an apple to the orbits of the planets […] Though it gave the right answer, using calculus was as much an act of faith as declaring a belief in God.”

Seife indicates that by radically embracing zero, calculus benefited from its revelatory aspects. In doing so, Seife again implies that zero, like God, is a transcendent source of knowledge not entirely reducible to human understanding.

“When you have infinity in an expression, or when you divide by zero, all the mathematical operations—even those as simple as addition, subtraction, multiplication, and division—go out the window. Nothing makes sense any longer.”

This is a simple explanation of the peril of zero. Zero and infinity cause even the most basic and indispensable aspects of mathematics to fail. No other number makes such a mess.

“Zero and infinity are two sides of the same coin—equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers. The troublesome nature of zero lies with the strange powers of the infinite, and it is possible to understand the infinite by studying zero.”

Seife again emphasizes the dualism of zero and infinity with an allusion to yin and yang, associating both zero and infinity with Eastern thought. His description of zero and infinity as “adversaries” is significant both for the way in which it recalls dualistic understandings of good and evil (e.g., Zoroastrianism) and because it implies that the relationship between zero and infinity is not always harmonious. Though linked, the concepts may also be in tension with one another.

“Zero and infinity are eternally locked in a struggle to engulf all the numbers. Like a Manichaean nightmare, the two sit on opposite poles of the number sphere, sucking numbers in like tiny black holes.”

This is a good example of Seife’s preference for looser, suggestive language over technical, precise language. He refers to Manichaeism, a dualistic philosophy judged heretical in the West and mentioned only once elsewhere in the book, thereby emphasizing the dualism of zero and infinity. He also foreshadows his examination of black holes as supreme instances of zero’s power.

“In Cantor’s mind there were an infinite number of infinities […] At the top of the chain sits the ultimate infinity that engulfs all other infinities: God, the infinity that defies all comprehension.”

Seife takes opportunities to mention theological applications of the ideas of zero and infinity because his book is about zero in every sense, not just the numerical one. He readily gives zero and infinity transcendental connotations where admissible; Cantor, for instance, was not only a mathematician but also a devout Lutheran.

“[I]nfinity and zero are inseparable and are essential to mathematics. Mathematicians had no choice but to learn to live with them. […] As mathematicians were uncovering the connection between zero and infinity, physicists began to encounter zeros in the natural world; zero crossed over from mathematics to physics.”

Seife could hardly convince readers that zero threatens entire cultures and philosophies if it only troubled a few geniuses pondering intangible mathematical constructs. The fact that zero directly influences how the world works frames it as immediately significant to readers.

“A zero in quantum mechanics means that the entire universe—including the vacuum—is filled with an infinite amount of energy: the zero-point energy. This, in turn, leads to the most bizarre zero in the universe: the phantom force of nothing.”

Seife maintains ordinary diction instead of diving into scientific jargon. Seife does not simply mention zero-point energy and move on; he gives it plenty of attention through the remainder of the book because it highlights the dualism of zero and infinity.

“[Z]ero is too powerful even for nature. When Einstein extended the theory of relativity to include gravity, he did not suspect that his new equations—the general theory of relativity—would describe the ultimate zero and the worst infinity of them all: the black hole.”

Seife emphasizes the black hole as a terrifying demonstration of zero’s danger. These unfathomably massive, light-devouring, almost immortal collapsed stars are dangerous yet mysterious and defy comprehension, just like zero and infinity.

“Zero is so powerful because it unhinges the laws of physics. It is at the zero hour of the Big Bang and the ground zero of the black hole that the mathematical equations that describe our world stop making sense. However, zero cannot be ignored. Not only does zero hold the secret to our existence, it will also be responsible for the end of the universe.”

Again emphasizing zero’s chaotic properties, Seife nevertheless positions it at the literal center of the universe via the Big Bang. His suggestion that it will also be central to the universe’s end builds on the book’s mystical overtones by recalling descriptions of God as both the beginning and the end (e.g., the alpha and omega of Christian symbolism). As it turns out, this end is one of infinite expansion and therefore not really an “end” at all, but rather a fate that underscores the tight association between zero and infinity.