Mystery Of The Aleph: Mathematics, The Kabbalah, And The Human Mind [Hardcover]

The search for infinity, that sublime and barely comprehensible mystery, has exercised both mathematicians and theologians over many generations: Jewish mystics in particular laboured with elaborate numerological schema to imagine the pure nothingness of infinity, while scientists such as Galileo, the great astronomer, and Georg Cantor, the inventor of modern set theory (as well as a gifted Shakespeare scholar), brought their training to bear on the unimaginable infinitude of numbers and of space, seeking the key to the universe.

In this sometimes technical but always accessible narrative, Amir Aczel, the author of the spirited study Fermat's Last Theorem, contemplates such matters as the Greek philosopher Zeno's several paradoxes; the curious careers of defrocked priests, (literal) mad scientists, and sober scholars whose work helped untangle some of those paradoxes; and the conundrums that modern mathematics has substituted for the puzzles of yore. To negotiate some of those enigmas requires a belief not unlike faith, Aczel hints, noting, "We may find it hard to believe that an elegant and seemingly very simple system of numbers and operations such as addition and multiplication--elements so intuitive that children learn them in school--should be fraught with holes and logical hurdles." Hard to believe, indeed. Aczel's book makes for a fine and fun exercise in brain stretching while providing a learned survey of the regions at which science and religion meet. --Gregory McNamee

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 infinityAof 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 mathAcalled set theoryA"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 mathAeven, 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.
Copyright 2000 Reed Business Information, Inc.

Adult/High School-Aczel tells of mathematicians struggling with absolute infinity and some of its mind-bending ramifications. The crown jewel of this struggle was conceived more than a century ago by Georg Cantor and remains an enigma to mathematicians. Cantor spent his life going back and forth between trying to prove and disprove his continuum hypothesis. In the Kabbalah, the aleph "represents the infinite nature, and the oneness, of God." Cantor deliberately picked this symbol for use in his equations: to him, trying to understand the absolute infinite was like trying to touch the face of God. About 50 years after his death, another mathematician definitively showed that the continuum hypothesis cannot be proven valid or invalid by any known means. Aczel provides a good history leading up to and past Cantor's work. Personal stories of people such as Pythagoras, Galileo, Newton, and G”del are mixed in with well-put explanations of the concepts they pondered. A brief history of the Kabbalah and highlights of some of its concepts help readers understand Cantor's work. The author writes cleanly and clearly on a complex subject, and readers don't have to be good at math to enjoy this book. It's perfect for analytically minded students who love to ponder big questions. Those who enjoyed the popular cosmology books by Stephen Hawking are likely to devour this one as well.

Sheila Shoup, Fairfax County Public Library, VA

Copyright 2001 Cahners Business Information, Inc.

If, as G. K. Chesterton once proposed, insanity constitutes the modern form of heresy, then Georg Cantor deserves recognition as one of modernity's supreme heretics, one who lost his sanity in challenging the limits of mathematical rationality. In this engrossing story of a man and of an idea, Aczel elevates Cantor out of obscurity into his proper place in cultural history, while confronting readers with the intellectual riddle that unhinged Cantor's powerful mind: the riddle of infinity. Discovered as a potent geometrical tool by the ancient Greeks, contemplated as a divine mystery by the medieval Kabbalists, infinity remained a mere potentiality for Newton and Leibniz, but it loomed above Cantor as a terrifying yet irresistible reality. In assessing Cantor's achievement as the first to probe infinity with mathematical rigor, Aczel demonstrates the same gift for interpreting complex concepts that he previously demonstrated in God's Equation [BKL S 15 99], about Einstein's pioneering work in cosmology. And as in his book on Einstein, Aczel penetrates to the human drama behind the formulas, detailing the personal frustrations and professional conflicts that drove Cantor into mental collapse. Aczel also uncovers the uncanny ways in which Cantor's life foreshadowed that of his more famous successor, Godel, who was attracted to the same problems and doomed to the same descent into madness. An indispensable book for anyone interested in the darker side of intellectual progress. Bryce Christensen
Copyright © American Library Association. All rights reserved

"...Aczel...serves as an expert guide on Cantor's voyage to the far reaches of abstraction." -- The Sciences, Nov./Dec. 2000

"An indispensable book for anyone interested in the darker side of intellectual progress." -- ALA Booklist (starred review), 10/01/2000

(Four Walls Eight Windows) The story of Georg Cantor, a brilliant mathematician who developed an understanding of the concept of infinity. Dicusses the contributions to Cantor's work and the philosophical aspects of infinity as Cantor saw it. DLC: Infinite.

Amir D. Aczel is the bestselling author of ten books, including Entanglement, The Riddle of the Compass, The Mystery of the Aleph, and Fermat's Last Theorem. He lives in Brookline, Massachusetts. --This text refers to the Paperback edition.