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

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.

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"

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.