The Big O
The Zero in Math—Why Nothing Matters
The discovery of zero—apparently the brainchild of several cultures worldwide—gave mathematicians not just a counting aid but a means to far more complex calculation.
Amir Aczel’s new book desperately wants to be an adventure story. He describes his travels as an “odyssey” and a “great adventure,” and compares himself to Indiana Jones. He imagines that a taciturn taxi driver in Vietnam might rob or kill him. (Nothing happens). He favors breathless rhetorical questions that suggest danger and daring. When he closes one chapter with the ominous query—“Could I rediscover it?”—you can almost sense the hostile natives hiding in dense jungle with poisoned blow darts trained on his head.
The key part of his question is the prefix “re.” Aczel is not discovering something new; as he freely admits, he’s searching for an inscription that was already discovered in 1891 and translated in 1931. The inscription contained a small dot of great importance in the history of ideas—it was the oldest extant zero. (Well, except for the Mayans and arguably the Babylonians.) When he finds the artifact with this inscription languishing in a Cambodian warehouse, he struggles to express the staggering significance of his own work: “This is the Holy Grail of all mathematics, I thought. And I found it.”
Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers is a strange and frustrating book. How the concept of numbers—and zero in particular—originated and developed is a fascinating topic rich with philosophical and practical implications. Aczel intermittently engages with these themes in a meaningful way, but much of the book is clogged with irrelevant and grandiose first-person material. He crisscrosses large swaths of Southeast Asia, but the most perilous thing that happens outside of his imagination is a routine shakedown by a customs official looking for a bribe. It doesn’t help that his idea of travel writing involves prose that would fit nicely in a botanical encyclopedia. (“later on the ground I identified ficus and giant tualang and dipterocarp, which can reach 200 feet and are often covered with lichens.”)
Aczel seems to fear that if he does not devote sufficient space to describing the peasants, snake charmers, and exotic flora he encounters, his readers might recoil in horror once they realize his subject is mathematical and philosophical. An obsession with the sensory is an odd quality in a book on the origins of numbers—concepts that are abstractions removed from the physical world. The moment when some humans first realized that three oranges, three dogs, and three fish all share some property of “Threeness” would not make for riveting cinema. It was an intellectual achievement: a recognition of a new way to understand the world. With the right guide, considering the nuances and implications of number theory and its history would be sufficiently interesting.
To appreciate why abstracting the concept of numbers away from particular things is an achievement, try to imagine a world without such a concept. One piece of evidence that might come from a pre-numerical world is the fibula of a baboon from roughly 20,000 years ago. It’s marked with a series of vertical notches that appear to represent early evidence of counting. Two human ancestors comparing tallies on baboon bones would probably have a rough sense of magnitude: if every hunter in the tribe made a notch for each animal he killed, the best hunter could display a bone with more notches.
But a sense of relative magnitude would not necessarily require a concept of numbers. You might know that the chief had more kills than anyone else without knowing precisely how many he had. We might even extend the paleo-fantasy and imagine a tricky rival fooling people by subtly increasing the space between each mark, thus creating the appearance of a greater number of marks (college students still use this tactic by adjusting the spacing and margins on papers).
The baboon fibula is not evidence of a number system, but it’s a preliminary step in that direction. A more efficient system uses a finite set of discrete symbols to represent an infinite set of numbers. Expressing the number 48 with two different symbols “4” and “8” is more efficient than writing a vertical dash 48 times, but it is not inevitable that we express 48 in the way that we do. In a base-12 system, for instance, the number 48 would be written “40.” (Instead of the “4” representing 4 tens, it would represent 4 twelves.) In a base-five system, the number 48 would be written as “143,” and in binary, the base-two system that drives modern computing, it would be written as “110000.”
Though inconvenient, it’s also possible to continue inventing differently shaped squiggles to symbolize numbers forever. The Romans did not express the numbers 50, 100, 500, and 1000 by recombining elements from a small set of symbols as we do; they used a new symbol for each of these larger numbers. L is 50, C is 100, D is 500 and M is 1,000. Zero, by contrast, allows the same symbols to cycle through different places in a notation system in order to represent increasingly large numbers without inventing new symbols.
Another option would be not to use symbols at all but simply to refer to familiar features of the world that are already linked to particular numbers. Under this model, you would walk into a bakery and ask for “eyes” loaves of bread. Aczel paraphrases an archaeologist who argued that in many ancient Southeast Asian cultures you might express the numbers 6, 5, and 4 by saying something like this: “the flavors, the organs of sense, and the Veda…there were six known flavors of food, five senses and four Veda (the ancient collections of Hindu holy writings).”
Zero was a crucial discovery in part because it allows for ease and economy of notation. We can distinguish at a glance, for instance, the numbers 14 and 104. We can also add and multiply these two numbers with much greater ease than a cumbersome system like Roman numerals would allow. Recognizing that zero was not merely a placeholder but also possessed mathematical properties itself helped make possible an expanded number system that includes imaginary, complex, and negative numbers.
Aczel has some intriguing speculations on how Buddhist concepts of emptiness might have given rise to zero. But he’s too quick to appeal to embarrassing clichés about how only an “Eastern mind” possesses the “very different kind of logic” required to invent the idea of zero. He also has a bizarrely competitive desire to prove that people in the East were “first.” In fact symbols that represent zero probably arose independently in Babylonian, Indian, and Mayan cultures. Like the alphabetic principle and the wheel, the idea of zero seems to be something that multiple human minds steeped in various beliefs and located in diverse places could discover.
Aczel seems unaware of any irony when he declares that he wants to find the first zero in order to “see it with my own eyes, to touch it, to feel it.” Conceiving of zero is an achievement because it requires us to transcend the sensory and realize that even if we can’t see or touch something, that doesn’t necessarily mean it isn’t there.