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"