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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

by Amir D. Aczel

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In the late nineteenth century, an extraordinary mathematician languished in an asylum. His life's work on "the continuum problem" would bring us closer than any mathematician before him in helping us understand the nature of infinity. This is the story of Georg Cantor.

Cantor's work, though brilliant, seemed to move in half-steps. The closer he came to the


In the late nineteenth century, an extraordinary mathematician languished in an asylum. His life's work on "the continuum problem" would bring us closer than any mathematician before him in helping us understand the nature of infinity. This is the story of Georg Cantor.

Cantor's work, though brilliant, seemed to move in half-steps. The closer he came to the answers he sought, the further away they seemed. Eventually it drove him mad, as it had mathematicians before him.

A respected mathematician himself, Amir D. Aczel follows Cantor's life and traces the roots of his deeply philosophical theories. From the Pythagoreans, the Greek cult of mathematics, to the mystical Jewish numerology found in the Kabbalah, The Mystery of the Aleph follows the search for an answer that may never truly be reached.

Editorial Reviews

The author of Fermat's Last Theorem has again transformed supposedly dull mathematics into appealing narrative. In The Mystery of the Aleph, he tells the story of Georg Cantor (1845-1918), a Russian-born German whose unprecedented research on "the continuum problem" touched on the very nature of infinity. But did Cantor's advanced work also imperil his sanity?
Washington Post
An engaging, pellucid explanation of the mathematical understanding of infinity, enlivened by a historical gloss of the age-old affinities between religious and secular conceptions of the infinite.
Publishers Weekly - Publisher's Weekly
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.) Copyright 2000 Cahners Business Information.|
School Library Journal
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.
Mystery of The Aleph provides intriguing insights into Jewish mysticism and math, drawing some unusual links between the story of Georg Cantor, a mathematician who languished in an asylum, and the theories which eventually began to be proved. Cantor's theory of the infinite holds many seeming contradictions; but its philosophical and spiritual roots have eventually proved sound - and it's the rest of mathematics which may be questioned. An intriguing blend of biography, spiritual insight and science.
Richard Bernstein
Mr. Aczel is very good at portraying the essences of the thoughts and lives of that quirky class of geniuses known as mathematicians, and he goes back not just to Zeno but to such other fascinating pioneers of number theory as Pythagoras and Eudoxus to do so. He deepens our appreciation of their discoveries by linking them to the equally deep, nonmathematical musings of the Kabalists, for whom infinity was a mystical equivalent to the immensity of God...Mr. Aczel's book remains highly enjoyable and frustrating at the same time. It deals, after all, with great minds venturing into the farthest reaches of speculation, with the nature of endlessness itself, both mathematical and religious, subjects that were not meant to be easy.
New York Times Book Review

Product Details

Sterling Publishing
Publication date:
Product dimensions:
6.00(w) x 8.16(h) x 1.00(d)

Read an Excerpt

0 Halle

By all rights, Halle should have held some attraction for Cantor, as his family members on both sides were gifted musicians. Some of them had achieved renown in their native Russia. But Cantor was not interested in the charms of Halle. His was a family of immigrants-from the Iberian Peninsula via Denmark and Russia-and young Cantor was pushed to excel. His father, in particular, sent Georg letters throughout the years urging him to do well at school and to live up to the great expectations of his family.

Halle is situated halfway between two great university cities: Berlin to the northeast and Gottingen to the west. During the late nineteenth century, the University of Berlin was the world's best in mathematics, and Berlin was one of the most vibrant and exciting cities in all Europe. Gottingen was the other academic magnet. Like Halle, Gottingen is an old medieval city. Many houses in the town center bear plaques with the names of famous former residents, from Heine the poet to Bunsen the chemist to Olbers the astronomer, and many others, most notable among them Carl Friedrich Gauss (1 777-i 8 5 5 ), arguably the greatest mathematician of the time. Cantor felt the pull of both Berlin and Gottingen.

But Cantor stayed in Halle, waiting for the invitation that never came. Over the years, whenever a mathematics opening became available at Berlin or Gottingen he pinned his hopes on it, and when he wasn't offered the position, he would go into a fit of rage. He had an intense, demanding personality, and an explosive nature. These attributes made him enemies and lost him friends throughout his life. In contrast with his behavior with other mathematicians, Cantorexhibited tenderness in his relationships with his family memhers. While he always dominated conversations with colleagues, at home Cantor took a more relaxed role, letting his wife and children initiate and lead conversations at the dinner table. He ended every meal by asking his wife: "Have you been pleased with me today, and do you love me?"

Cantor started as a Privatdozent, the entry-level academic job at German universities of the time. Within a few years of hard work, he was promoted to Associate Professor, and shortly afterwards a Professor of Mathematics. Cantor became involved in intensive research in mathematics, but in the midst of his most productive period, something strange happened, which put a temporary end to his work. In the summer of 1884, Georg Cantor was struck by deep depression. From May through June of that year he was immobilized-unable to work or do much of anything. His condition distressed his wife and children and perplexed his colleagues, who saw in him a mathematician aspiring to great heights. However, without any professional help or medication, Cantor recovered from his illness and returned to normal life. Afterwards, he wrote a letter to a close friend, the Swedish mathematician Gosta Mittag-Leffler (1846-1927), describing his illness and mentioning that just before the mental breakdown he was working on the "continuum problem."

The following year, 1885, Cantor built an opulent house for his family on Handelstrasse, a street named after Halle's great composer. The house is still owned by Cantor's grandson. It is a two-story building, with high ceilings and tall windows. Georg Cantor's father, a merchant and stockbroker, had died a few years earlier, leaving his heirs half a million marks. Some of the inheritance money went into building the new house and buying furnishings so that the Cantor family could live in comfort...

Meet the Author

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.

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