Zero: The Biography of a Dangerous Idea

These chapters discuss zero’s breakthroughs—how it finally overcame the stronghold of Aristotelianism and how mathematicians finally learned to accommodate it in their systems—and the resulting benefits. Seife outlines the West’s difficult and contentious transition from avoiding the void to realizing that zero, though nothing, is useful for some things. By the end of Chapter 6, Seife grants that the West, conquered by zero, had eventually conquered zero, though this triumph would not last.

With its focus on the West’s discovery of zero’s unexpected utility, this part emphasizes The Revelation of Zero—how accepting zero granted various individuals access to the mysteries of the universe, if at the cost of paradoxes and accusations of heresy. Brunelleschi’s application of the vanishing point (a zero-dimensional point at infinite distance), Pascal’s discovery of the vacuum, the solving of Zeno’s paradox, and the multifaceted invention of the Riemann sphere, which reunited shapes and numbers and revealed The Dualism of Zero and Infinity, are examples. Seife is especially anxious to impress upon readers how integral zero was to the discovery of calculus and how useful calculus was. Seife strongly suggests, partly via his choice of subtitles, that calculus was initially a “mystical” art, bypassing reason by faith because it depended upon the illogical operations of dividing by zero and summing infinite zeros to get a sum greater than zero. He repeatedly states that mathematicians willingly embraced calculus despite its problems because it revealed how nature works. Zero, like an unexpected vision from above, revealed hidden truths despite remaining mysterious itself.

Seife’s expansive definition of zero remains evident in these chapters. The vanishing point, for instance, most commonly features in discussions of art, yet Seife treats it as a manifestation of zero, much as he later will scientific concepts including absolute zero (the lowest temperature), zero-point energy, ground zero (the singularity of a black hole), and zero hour (the instant of the Big Bang). The breadth of Seife’s conception of zero points both to the number’s relationship to infinity—zero seems, paradoxically, to be everything even as it is nothing—and to its centrality to Seife’s thought.

Nevertheless, it’s the zero of mathematics that dominates Seife’s discussion in these chapters. The equations and mathematical constructs in these chapters are the most difficult in the book, so Seife guides readers step by step and includes more graphs to illustrate the abstract concepts. Readers must keep up with the mathematical discussions to appreciate what Seife frames as a crowning achievement: the Riemann sphere, which elegantly and cleverly synthesized many seemingly disparate mathematical ideas.

Seife’s tone also aims to help readers through the subject matter. Though objective and informative in recounting the facts, Seife’s discussion often evinces wonder and curiosity as he explores some of the most esoteric realms of mathematics, where dwell irrational and imaginary numbers, infinitesimals and limits, infinite sums and infinite sets, and even different kinds of infinity. This last point is especially paradoxical; if infinity extends endlessly, it is not immediately obvious how there could be smaller or larger “infinities,” as in the case of Cantor’s sets. This counter-intuitiveness is a property infinity shares with zero, and it is this strangeness rather than the mathematical mechanics that Seife wants readers to walk away with.