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COPYRIGHT 2003 All rights reserved. Reproduced by permission of The Condé Nast Publications Inc.
Few ideas have had a racier history than the idea of infinity. It arose amid ancient paradoxes, proceeded to baffle philosophers for a couple of millennia, and then, by a daring feat of intellect, was finally made to yield its secrets in the late nineteenth century--though not without leaving a new batch of paradoxes. You don't need any specialized knowledge to follow the plot: the main discoveries, despite the ingenuity behind them, can be conveyed with a few strokes of a pen on a cocktail napkin. All of this makes infinity irresistible meat for the popularizer, and quite a few books in that vein have appeared over the years. Now, in "Everything and More: A Compact History of Infinity;" (Norton; $23.95), the celebrated author David Foster Wallace has set out to initiate readers into its mysteries.
It might seem odd that finite beings like us could come to know anything about infinity, given that we have no direct experience of it. Descartes thought that the idea of infinity was innate, but the behavior of children suggests otherwise; in one study, children in their early school years "reported 'counting and counting' in an attempt to find the last number, concluding there was none after much effort." As it happens, the man who did the most to capture infinity in a theory claimed that his insights were vouchsafed to him by God and ended his life in a mental asylum.
Broadly speaking, there are two versions of infinity. The woollier, more mystical one, which might be called metaphysical infinity, is associated with ideas like perfection, the absolute, and God. The more hardheaded version, mathematical infinity, is the one that Wallace's book is concerned with. It derives from the idea of endlessness: numbers that can be generated inexhaustibly, time that goes on forever, space that can be subdivided without limit. While metaphysical infinity tends to evoke awe in those who contemplate it, mathematical infinity has, for most of Western intellectual history, been an object of grave suspicion, even scorn. It first cropped up in the fifth century B.C., in the paradoxes of Zeno of Elea. If space is infinitely divisible, Zeno argued, then swift Achilles could never overtake the tortoise: each time he caught up to where the tortoise was, it would have advanced a little farther, ad infinitum. So traumatizing were such paradoxes that Aristotle was moved to ban the idea of a "completed" infinity from Greek thought, setting the orthodoxy for the next two thousand years.
The eventual rehabilitation of the infinite had its origins in another...
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