Customer Reviews

The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind

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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.

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.

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.

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.

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.

This is a very enjoyable book about infinity and its history. It reviews the genesis of the concept of infinity from early times (ancient Greece), through the Middle Ages and on to the modern era. The author carefully surveys the major developments in this history, including the works of Galileo, Bolzano, Cantor, Godel, and others. He adds interesting asides about number mysticism both in the Jewish Kabbalah and in Christian teachings. The mathematics is kept at a low, understandable level, and the biographies of the major figures in this 2,500 year old drama of human understanding are very well developed. Overall, this is a readable and informative book. My only wish as a reader would have been to have more space devoted to the psychology of the actors in this drama. Many of these mathematicians went crazy. The question in every reader's mind is: Why?

This is easily the best book on mathematics this year. Amir Aczel has done it again, after Fermat's Last Theorem and God's Equation. Here he tackles one of the most difficult areas in mathematics--set theory--and weaves a very readable narrative including elements of Jewish mysticism and psychology. This book deals with the tormented life of Georg Cantor, the first person in history to understand the nature of infinity. Read it! I will say no more, so I don't spoil your enjoyment.

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 do not however wish to suggest that the mathematics involved makes the book inaccessable to some, quite the opposite is true.Personally, I would reccomend that one take the neccesary time to think certain things through rather than read on immiedietly, in order to try come to terms with a concept that "defies intuition". A great compliment to this book,is "To Infinity And Beyond" by Eli Maor which looks at infinity in various contexts. Many concepts that are raised in "The Mystery of The Aleph" are discussed in more detail, while retaining the reader's interest and fascination.

The theme of this book seems to be to connect contemplation of infinity with insanity. The Kabbalah connection seems weak, but the mathematics of the infinite gets pretty good treatment. Georg Cantor (mid 1800s)the discoverer of the infinite series of infinite numbers aleph0, aleph1, aleph2, etc. was mentally unbalanced. So it seems was Kurt Godel who discovered (1930s) some amazing things about the limits of mathematical, and by extension, human knowledge.

But why attribute Cantor's and Godel's mental instability to the contemplation of the infinite? Lots of other mathematicians have contemplated the infinite without seeming to go mad. Paul Cohen, for one, figures in this tale for a discovery that dates to 1964.

Lots of space is given over to the continuum hypothesis that 2 raised to the aleph0 equals aleph1. The continuum hypotesis bedeviled Cantor and Godel. It wasn't until 1964 that Cohen showed that the continuum hypothesis is independent of the axioms for set theory, even the axioms for set theory with the axiom of choice thrown in.

We also now know that the axiom of choice is independent of the axioms for set theory. Geometry forked over the parallel axiom and it can be assumed to be true or false, or true in another form. You can assume there is none, one, or more than one line parallel to a given line, and passing through a point not on the given line.

Set theory is free to fork over the axiom of choice, or over the continuum hypthesis. Since the continuum hypothesis is independent of the axioms for set theory, you can safely (consistently) assume either it or its negation is true. You end up with two very different universes.

All and all this book is a little light especially in the author's treatment of Cohen's forcing method. It's pretty clear otherwise but we must still wait for other treatments of these and similar results. It's too bad because somehow the significance of what Cantor, Godel and Cohen discovered needs to be communicated to a broader audience. Others have tried - Godel, Escher and Back is one such title from a few years ago. It did not seem to pull it off either. If I was contemplating purchasing this book, I'd wait for the paperback copy.

As a professional mathematician I can tell you that Aczel has done an admirable job. He tackles in this book some of the thorniest issues in modern mathematics: the axiom of choice and the continuum hypotheis. Aczel's explanations of topics such as Russell's paradox, Zermello's set theory, and Godel's work are among the best I have seen. He tells the story of the continuum hypothesis with verve and style. I strongly recommend this book to anyone who is interested in the history of mathematics.