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

Amir D. Aczel
3.3 out of 5 stars  See all reviews (3 customer reviews)
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Book Description

1 Aug 2001 Mathematics, the Kabbalah and the Search for Infinity
A compelling narrative that blends a story of infinity with the tagic tale of a tormented and brilliant mathematician. From the end of the ninteenth century until his death, one of history's greatest mathematicians languished in an asylum, driven mad by an almost Faustian thirst for universal knowledge. THE MYSTERY OF THE ALEPH tells the story of Georg Cantor (1845-1918), a Russian born German whose work on the 'continuum problem' would bring us closer than any mathemetician before him in helping us to comprehend the nature of infinity. A respected mathematician himself, Dr. Aczel follows Cantor's life and traces the roots of his enigmatic 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 trusted.

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Product details

  • Paperback: 274 pages
  • Publisher: Washington Square Press; New edition edition (1 Aug 2001)
  • Language: English
  • ISBN-10: 0743422996
  • ISBN-13: 978-0743422994
  • Product Dimensions: 14 x 21.3 x 1.8 cm
  • Average Customer Review: 3.3 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Bestsellers Rank: 651,904 in Books (See Top 100 in Books)

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Amazon Review

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


"An engaging, pellucid explanation of the mathematical understanding of infinity, enlivened by a historical gloss on the age-old affinities between religious and secular conceptions of the infinite."

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On January 6, 1918, an emaciated and weary man died of heart failure at the Halle Nervenklinik, a university mental clinic in the German industrial city of Halle. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Customer Reviews

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Most Helpful Customer Reviews
7 of 7 people found the following review helpful
4.0 out of 5 stars Defies Intuition but it's well worth a try. 10 Jan 2001
By A Customer
Once again Amir Aczel has provided us with an enthusiastic and intriguing look at a fascinating subject.Trying to come to terms with the Infinite is a difficult task that, as explained in the book, has claimed many victims. Aczel does a wonderful job turning the reader into a victim, although thankfully not as critical as the likes of Cantor and Godel. Much like his previous books Aczel blends the science( in this case mind boggling mathematics)into a fascinating background, paying great attention to the lives and characters of those concerned. The biographies of Gallileo and Kurt Godel are particularly interesting, especially as one would think that there was little left to know about them. However the centre of attention is the life and work of Georg Cantor, the mathematician synonomous with discoveries concerning the infinite. It would be difficult to find a more interesting and bizarre story than that of Georg Cantor but the real source of intrigue are his ideas and those of others concerning the subject of infinity. Added to that is a touch of mystisism in the form of "Kabalah" making Infinity an even more awesome concept.
While the book is written in an entertaining and absorbing style with ideas explained simply and concisley, much contemplation is required by the reader. I personally would reccomend that one take the neccesary time in order to try come to terms with a concept that "defies intuition". Four stars then, and the only reason I didn't give it five is because I can still sleep at night...barely.
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4 of 5 people found the following review helpful
4.0 out of 5 stars Contemplating the Infinite 13 Mar 2004
By A Customer
Format:Paperback|Verified Purchase
One of only a handful of absorbing books that I have read in one sitting. It started extremely well so I was a little shaken when I read on page 24 how Archimedes compared the volumes of a sphere and a cone (it was a sphere and a cylinder I believe). However, rest assured, this was the only glitch I found in an excellent exposition of the Infinite from a set-theoretical approach.
As this book is not particularly mathematical, and is extremely well-written, it should appeal to those readers with a strong interest in mathematics, without the mathematical hardware behind them. Aczel's book makes a great accompaniment to Rudy Rucker's "Infinity and the Mind", which is much more mathematical, and covers many more aspects of Infinity. I give Aczel's book 4 stars, as Rucker's is out there on its own with 5 for this particular topic area.
To complete the trilogy I recommend Maor's "To Infinity and Beyond", which is also a very easy read without being too mathematical.
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2.0 out of 5 stars Warning: read the subtitle! 7 Jun 2013
By Mr R
Format:Paperback|Verified Purchase
Appearing below the title of this book is a reference to the Kabbalah, something which is referred to ad nauseum throughout this book. Glance through the index, where you will find dozens of references to God, the Kabbalah, Jewish mysticism, prayers, etc. Worse still, the page numbers given in the index represent only some of the references: in truth, the word 'God' permeates the text, and this is regrettable in a book purporting to be mainly about mathematics. Here's a sample (pp145-146): "...a verse of this prayer, recited by Jews several times daily, is that God rules the universe [there follows a sentence in Hebrew]. Hence the concept of infinity was known to anyone with a Jewish background". Or on page 156 we read, "Was Cantor's fate very different from those of the rabbis of the second century who tried to enter God's secret garden...?" These examples are not the worst excesses: they are typical.

There is some neatly explained mathematics, however, and it is only fair to say how entertainingly Aczel writes: he makes the subject accessible to non-mathematicians, and in this respect provides a great service. Chapters 3 and 20 are a joy to read. There are one or two careless errors though: for example, on page 24 we read, ".. the volume of a cone inscribed in a sphere with maximal base equals a third of the volume of the sphere" [in fact it's a quarter]. Again, on page 155, Aczel uses the word 'converse' when 'negation' would be appropriate.
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Amazon.com: 3.8 out of 5 stars  62 reviews
42 of 42 people found the following review helpful
4.0 out of 5 stars mathematics, cantor and mysticism 24 Jan 2008
By Michael R. Chernick - Published on Amazon.com
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.

Aczel also discusses the lives of Galileo and Bolzano and their contributions to mathematics. I was aware of the one-to-one correspondence between the integers and the square of the integers. The fact that the discovery goes back to Galileo was news to me. While I knew of Galileo for his invention of useful telescopes and his contributions to astronomy, I had no idea that he had made such a fundamental contribution to mathematics.

As with some of the other reviewers, I find the discussion of the Kabbalah somewhat weak and perhaps misplaced. I also think there is a mathematical error in this chapter. Aczel states that there are 10 permutations of the arrangement of the Hebrew name for God, YHVH, and he places importance on the number 10. He enumerates the permutations to be YHVH, YVHH, VYHH, VHYH, HVYH, HYVH, HVHY, HYHV, HHYV AND HHVY. This puzzled me. As I thought about my combinatorial mathematics I thought the correct answer should be 12. I tried a complete enumeration myself and found 12. It seems that Aczel missed YHHV and VHHY.

Aside from this, the discussion of mathematics is generally good. It is not detailed and is written in a popular style to be readible to a general audience. The heart of the book is the life of Georg Cantor. Cantor aided by the work of Galileo and Bolzano and his teacher Karl Weierstrass made the breakthroughs that led to the development of transfinite numbers and modern set theory. He worked mostly in isolation at Halle University and was frustrated by never being granted an appointment at University in Berlin where most of the famous mathematicians of the time resided. His conflict with Kronecker is discussed and the support he got from Mittag-Leffler is also covered.

Aczel provides background to varying degrees on all the mathematicians that he discusses and we feel that we understand their personalities and the underlying reasons for the positions that they took. Cantor's bouts with insanity are also described. Although it could be simply that he was suffering from manic depression (a disorder that was not understood at the time), Aczel attributes Cantor's insanity to the frustration of his efforts to cope with infinity. Certainly there must have been frustration over his inability to prove the continuum hypothesis (later determined to be unprovable) and the lack of universal acceptance of his ideas in the mathematical community.

However, I agree with some of the other reviewers who think that Aczel's thesis, that doing mathematical research on infinity might induce insanity, is a bit farfetched. In covering the life of Kurt Godel, a important successor to Cantor, Aczel points to Godel's bouts with insanity to try to reinforce this thesis. Godel did not have the same issues in his life history that Cantor had. Still, other mathematicians that worked in this area including Russell and Cohen never had similar bouts.

Coverage of the work of Godel and Cohen brings the reader up to the current state of knowledge about transfinite numbers and set theory. For the mathematically inclined there is an appendix at the end that provides statements of Zermelo's axioms that are the basis of modern set theory. It is within this system that the axiom of choice and the continuum hypothesis are both consistent and independent and therefore can neither be proven to be true or false.

If you like reading about the history of mathematics and the personalities of important mathematicians you will enjoy this book inspite of a few flaws.
49 of 52 people found the following review helpful
4.0 out of 5 stars A somewhat flawed, magical, fascinating read 24 April 2002
By Dennis Littrell - Published on Amazon.com
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? Beyond the infinities might there exist something more "irrational" more completely "transcendent" than we can imagine even in our wildest fantasies?

At any rate, reading Aczel's magical book, I am persuaded to think so. And I can understand how New Agers and Kabbalists can become so enamored of numbers that they slip quite imperceptibly into numerology. (Numerology being to mathematics what astrology is to astronomy.)

Where I think Aczel is off the mark is in suggesting that it was concentration on the continuum that led to the ill mental health of Georg Cantor and Kurt Gödel. The old saw about thinking so long and hard on a subject leading to madness is something however that won't go away. In chess we have the preeminent examples of Paul Morphy and Bobby Fischer, both towering genius like Cantor and Gödel, who slipped into delusion and paranoia after plummeting the depths of Caissa. With the great strides being made in neuroscience today, we might one day understand what these men had in common besides great intelligence and the ability to concentrate to an extraordinary degree.

There is a lot of interesting material throughout the book. I was especially intrigued with an implication of the fact that an infinite number of steps (e.g., 1/2 + 1/4 + 1/8...etc.--convergence) could lead to a finite sum. (p. 12) This really implies to my mind that we can relate in some sense to the idea of infinity. I contrasted this with Aczel's assertion on page 90 that if one could choose at random a number on the real line, that number would be "transcendental with a probability of one" (missing by force any of an infinity of rational numbers). However, as Aczel points out elsewhere, one cannot actually choose a number randomly out of an infinite collection!

I also liked the report about the exasperated Paris Academy in the nineteenth century passing "a law stating that purported solutions to the ancient problem" of squaring the circle "would no longer be read by members of the academy." (p. 89) This reminded me of the action by the U.S. Patent Office some many years ago of refusing to accept patent applications for perpetual motion machines.

Aczel gives Cantor's proof of a higher order of infinity for transcendental numbers on page 115. It is a very beautiful proof that can be understood with very little knowledge of math. On page 112 he gives Cantor's equally beautiful proof that rational numbers are as infinite as whole numbers. However his gloss at the top of the next page I think contains some typographical error that makes it not helpful. But perhaps I am wrong. (Maybe somebody knows and would tell me.) There is also some confusion about when Gödel married Adele on pages 198 and 200, and there are perhaps too many typos in the book, e.g., on the first sentence of page 162 the word "of" is missing, and on page 164 the word "way" (or something similar) should follow the word "humiliating." Also note Michael R. Chernick's correction in his review below showing the two missing permutations for the Hebrew word for God that Aczel left out on page 32.

Despite these flaws, this is overall an extremely engaging book and a delight to read.

--Dennis Littrell, author of "The World Is Not as We Think It Is"
30 of 31 people found the following review helpful
3.0 out of 5 stars Finite, all too finite 28 Dec 2002
By John S. Ryan - Published on Amazon.com
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.
21 of 23 people found the following review helpful
5.0 out of 5 stars Infinite Understanding 23 Oct 2000
By R. Hardy - Published on Amazon.com
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. This would seem to indicate that all infinities are the same size; however, Cantor showed that this was not true. Specifically, he showed that although the rational numbers could be paired up with the counting numbers, there were not sufficient pairs to be made if you included such numbers as the transcendental irrationals pi or e.
Cantor went mad, and died in a psychiatric hospital; it is too much to say that contemplating infinities made him crazy, but his continued attempts to prove his Continuum Hypothesis provided increasing frustration, as did attacks from his fellow mathematicians. Gödel himself showed that the continuum hypothesis could not be proved or disproved. During his work on this problem of infinities, he began to go mad as well, showing his own symptoms of paranoia and obsessiveness. Eventually, he was convinced that his food was poisoned and he would touch less and less of it; he simply starved himself to death.
So open up these pages if you dare; studying infinities has not been healthful for everyone. Aczel, however, does not go deeply into proofs, using good illustrations to provide access to non-mathematicians for some distinctly strange mathematical ideas.
17 of 18 people found the following review helpful
5.0 out of 5 stars Aczel writes another winner 5 Dec 2000
By Peter D. Mark - Published on Amazon.com
Mr. Aczel's new volume on Cantor artfully weaves mathematics, history, religion, and psychology into a coherent narrative. The organizing theme is the historical development of the concept of infinity. Aczel traces infinity's history from the Pythagoreans of Classical Greece through the work of modern logicians and mathematicians such as Godel and Cohen, focusing on the contributions of Georg Cantor.
Aczel gives admirably pithy biographical summaries of the main players in this drama, including Galileo, Bolzano, Weierstrass, Kronecker, and Dedekind, and he brings to life the evolution of the key ideas. Particularly striking is the intellectual battle between Cantor and his teacher Kronecker, whose fundamental philosophical differences concerning the nature of infinity degenerated into a bitter personal feud.
Aczel sensitively draws parallels between Cantor's investigations of infinity and the Kabbalistic explorations of the Jewish mystics. He notes the importance of Cantor's and Godel's work on Turing's formal description and investigation of computation in the 1930s, but could have given more detail on how Turing used Cantor's diagonalization argument to show that uncomputable functions exist and that such problems as the Halting Problem are undecidable. This is a minor quibble. Overall, Aczel has pulled off a real coup by giving an engaging account of a fascinating story combining intellectual history, spiritual exploration, and human drama.
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