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The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity Paperback – September 1, 2001

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Editorial Reviews Review

The search for infinity, that sublime and barely comprehensible mystery, has exercised both mathematicians and theologians over many generations. Jewish mystics, in particular, labored 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 Shakespearean 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, 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 where science and religion meet. --Gregory McNamee --This text refers to an out of print or unavailable edition of this title.

From Publishers 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 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. --This text refers to an out of print or unavailable edition of this title.

Product Details

  • Series: Mathematics, the Kabbalah and the Search for Infinity
  • Paperback: 272 pages
  • Publisher: Washington Square Press; New edition edition (September 1, 2001)
  • Language: English
  • ISBN-10: 0743422996
  • ISBN-13: 978-0743422994
  • Product Dimensions: 5.3 x 0.7 x 8.2 inches
  • Shipping Weight: 8.5 ounces (View shipping rates and policies)
  • Average Customer Review: 3.8 out of 5 stars  See all reviews (65 customer reviews)
  • Amazon Best Sellers Rank: #534,241 in Books (See Top 100 in Books)

More About the Author

Amir D. Aczel, Ph.D., is the author of 17 books on mathematics and science, some of which have been international bestsellers. Aczel has taught mathematics, statistics, and history of science at various universities, and was a visiting scholar at Harvard in 2005-2007. In 2004, Aczel was awarded a Guggenheim Fellowship. He is also the recipient of several teaching awards, and a grant from the American Institute of Physics to support the writing of two of his books. Aczel is currently a research fellow in the history of science at Boston University. The photo shows Amir D. Aczel inside the CMS detector of the Large Hadron Collider (LHC) at CERN, the international laboratory near Geneva, Switzerland, while there to research his new book, "Present at the Creation: The Story of CERN and the Large Hadron Collider"--which is about the search for the mysterious Higgs boson, the so-called "God particle," dark matter, dark energy, the mystery of antimatter, Supersymmetry, and hidden dimensions of spacetime.
See Amir D. Aczel's webpage:
Video on CERN and the Large Hadron Collider:

Customer Reviews

Most Helpful Customer Reviews

51 of 52 people found the following review helpful By Michael R. Chernick on January 24, 2008
Format: Hardcover
I started reading this book on the plane that took me to my new home in New Jersey. I finished it about a month later. I am a slow reader and I also was very busy getting settled into my new job. As I prepared to write my review for Amazon I looked at the many other reviews that had already been written and I found that they were quite mixed. Some raved about it and some hated it. There were many good points on both sides.
I hope my review adds something new for potential readers to think about.

I am a mathematician by training. I have a bachelor's degree in mathematics and also a masters degree. In my university education I learned about algebra and analysis and did have some acquaintance with the results of Cantor on transfinite numbers. I also knew some things about the axiom of choice, the continuum hypothesis and the Hahn-Banach theorem. I got this education in the late 1960s and early 1970s. In the mid 1970s I went on to Stanford where I studied Operations Research and Statistics eventually leading me to a career as a statistician. I had not given much thought to these mathematical ideas in a long time.

While at Stanford, I did hear about Paul Cohen who was then considered to be a star in the Mathematics Department because of his great discoveries in set theory and logic at an early age.

This book provided me with an interesting reminder of my past education and cleared up a few ideas in logic that had been puzzling to me.

At first I thought I was going to hear about the life story of Georg Cantor, the father of transfinite numbers. I was pleasantly surprised to find out that the book develops ideas about infinity and infinite numbers going back to the time of the Greeks and the discovery of irrational numbers by the Pythagorean school.
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49 of 52 people found the following review helpful By Dennis Littrell HALL OF FAMETOP 500 REVIEWERVINE VOICE on April 24, 2002
Format: Hardcover
Aczel's fascinating book is a short narrative history of the concept of infinity (the aleph) with a concentration on its mathematical development, especially through Galileo, Cantor, Gödel, Paul Cohen and others, meshed with some very interesting material from the ancient Greeks and the Kabbalists who associated infinity with their ideas of God. He includes some material on how strikingly difficult it was for Cantor and others to go against established ideas. I think it was also Aczel's intent to force the reader to think about infinity as something spiritual. At least his book had that effect on me.

God is infinity, the ancient Kabbalists proclaimed, and indeed an all-powerful, all-knowing, immovable yet irresistible God may be something akin to infinity. God is perhaps a higher order of infinity, above the infinity of the transcendental numbers: infinity to the infinite power, one might say, and having said that, one might dismiss it all from the mind as being hopelessly beyond all comprehension. Yet, here, Amir Aczel brings us back. Cantor showed that we can think about infinity, at least to the extent that we can prove differences among infinities. We can, as it were, and from a distance, make distinctions about something we cannot really grasp. In a sense it is similar to contemplating what is beyond the big bang, or imagining the world below the Planck limit. Our minds were not constructed to come to grips with such things, yet maybe we can know something indirectly.

Maybe. In science what we know is forever subject to revision; but in mathematics we are said to have eternal knowledge. When it is proven (barring error) it is settled. Still, might mathematics exist beyond even the furthest reach of the human mind with a higher order affecting our proofs?
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30 of 31 people found the following review helpful By John S. Ryan on December 28, 2002
Format: Paperback
I really expected to like this book. God, infinity, Kabbalah -- how could you _miss_?
It was okay, I guess.
So what happened? Well, frankly, although the biographical information on Georg Cantor and Kurt Goedel is pretty good and the mathematical history is reliable, there's no real meat in the discussions of either infinity or the Kabbalah. Every time I thought Aczel was really going to get rolling and make a profound connection, he sort of petered out and changed the subject.
It's too bad, because Aczel really does have an important point lurking in here: the mathematics of infinity really does provide a window into the Ein Sof, and there probably is a connection (both historical and deeper) between the Kabbalistic and the Cantorian uses of the Hebrew letter alef. I'd have enjoyed some more thorough exposition, even at an elementary level, of both sides of this equation.
But for that, the reader will have to look (for infinity) to Rudy Rucker's _Infinity and the Mind_ or (slightly more elementary) Eli Maor's _To Infinity and Beyond_, or (for the rest) to any of numerous sources on Kabbalah. This book is only about a quarter-inch deep.
On the plus side, though, I will say that this isn't a bad book for somebody who has never encountered the subject(s) before. Just don't expect a lot of specificity; Aczel usually doesn't offer much more than vague allusions.
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24 of 26 people found the following review helpful By Rob Hardy HALL OF FAMETOP 500 REVIEWER on October 23, 2000
Format: Hardcover
People have tried this for a few thousand years to understand the infinite, most along religious lines. _The Mystery of the Aleph_ (Four Walls Eight Windows) by Amir D. Aczel traces the history of these understandings, but concentrates on the mathematical understanding that was really begun only in the last century. Galileo contemplated two sets, the counting numbers 1, 2, 3, 4... and the square numbers 1, 4, 9, 16.... He found that every square from the second set could be paired with a number from the first: 1/1, 2/4, 3/9, 4/16, and so on. This means that although there is an infinity of numbers in either set, one set is exactly as big as the other. Galileo was shocked that this was true, even though it seems as if there are many more numbers in the first set; but he had found the key property of an infinite set, that it can be equal to a set included within itself. Bernhard Bolzano built on this strange finding to show that a line one inch long has as many points as a line two inches (or any number of inches) long.
Georg Cantor is the mathematician most identified with studying infinities. Aczel's book is pretty good at explaining his very peculiar findings. Cantor found, for instance, that the infinity of counting numbers could be placed in a one to one correspondence with fractions (rational numbers). Of course, the fractions are more dense, given all of them that exist between only, say, 1 and 2. But the number of such fractions does not exceed the number of counting numbers. Cantor also had clever demonstrations that a one inch line had just as many infinite points on it as a one inch square plane, as did any size line and any size plane; the same was true of higher dimensions as well.
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