cover image The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity

The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity

Amir D. Aczel. Basic Books, $24.95 (304pp) ISBN 978-1-56858-105-7

Aczel's compact and fascinating work of mathematical popularization uses the life and work of the German mathematician Georg Cantor (1845-1918) to describe the history of infinity--of human thought about boundlessly large numbers, sequences and sets. Aczel begins with the ancient Greeks, who made infinite series a basis for famous puzzles, and Jewish medieval mystics' system of thought (Kabbalah), which used sophisticated ideas to describe the attributes of the one and infinite God. Moving to 19th-century Germany, mathematician Aczel (Fermat's Last Theorem) introduces a cast of supporting characters along with the problems on which they worked. He then brings in Cantor, whose branch of math--called set theory--""leads invariably to great paradoxes,"" especially when the sets in question are infinite. Are there as (infinitely) many points on a line as there are inside a square or within a cube? Bizarrely, Cantor discovered, the answer is yes. But (as he also showed) some infinities are bigger than others. To distinguish them, Cantor used the Hebrew letter aleph: the number of whole numbers is aleph-null; the number of irrational numbers, aleph-one. These ""transfinite numbers"" pose new problems. One, called the continuum hypothesis, vexed Cantor for the rest of his life, through a series of breakdowns and delusions: others who pursued it have also gone mad. This hypothesis turns out to be neither provable, nor disprovable, within the existing foundations of mathematics: Aczel spends his last chapters explaining why. His biographical armatures, his clean prose and his asides about Jewish mysticism keep his book reader friendly. It's a good introduction to an amazing and sometimes baffling set of problems, suited to readers interested in math--even, or especially, if they lack training. B&w illustrations not seen by PW. 5-city author tour; $30,000 ad/promo; 30,000 first printing. (Sept.)