The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind
by Amir D. Aczel
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Hardcover
(15 January, 2000)
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Editorial 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 ... Read more Reviews (52)
Trying to hard to ride the Kabbalah trend
It is highly ironic that at the start of this book the author chides those celebrates who like to play Kabbalah just because of its new found trendyness because this is exactly what seems to be going on in this book.
While Aczek does an excellent job of looking at the history of the concept of infinity as well as providing a clear survey of the mathematical principles involved but then basically shoots himself in the foot by trying to shove the Kabbalah in where it doesn't belong.Aczek himself may have recognized this because despite putting it into the title he never really spends much time developing the connection in the book except through somewhat veiled references and hints (no doubt because he knew if he tried to show a connection more clearly it could be torn apart by anyone who read the book.)
I think that Janna Levin offered a much better explication for the seeming connection between theoretical mathematicians and madness in her book How the Universe Got Its Spots when she wrote "The lore is that their theories drove them mad, though I suspect they were just
lonely, isolated by what they knew."
This would have been a much better book without the unrelated Kabbalaistic elements tacked on - but then again it might not have sold as many copies (and one gets the impression that's why it was included in the first place.)
Good historical reasearch combined with original thinking
You might call this book "the history of the scientific quest for infinity." The author, Amir D. Aczel, traces the struggle to embrace infinity back to Ancient Greece.The Greeks in their quest encountered some troubling paradoxes, which they never quite resolved. As a result they tended to shy away from infinity. (More accurately, they allowed for an "uncompleted" or "potential" infinity, but not a "completed" infinity.)
The modern phase begins with Galileo, who observed that, in a sense, "There are just as many integers that are perfect squares (e.g., 4, 9)." This was a truly amazing discovery. What appears to be a small subset (of integers) is actually just as large (in a sense) as the original. Galileo however failed to develop this observation. Perhaps he was so stunned he did not know how to proceed. Besides -- he had significant personal problems with the Inquisition.
Following Galileo, many mathematicians revisited concepts involving infinity. As a rule they tended to follow the Greeks in not allowing a "completed infinity," but only an "uncompleted infinity." All this was to change with Georg Cantor.
Georg Cantor's work was truly amazing. Although his work has been largely incorporated into mainstream mathematics, the shock waves he produced were so great there are still rumblings to be heard. Perhaps most amazing about Cantor's theory is that there are "infinitely many infinities - each one larger than the previous." Furthermore, there is no largest.
In the history of mathematics there have been some surprisingly virulent feuds. Among the most extreme are: Cardano-Tartaglia, Newton-Leibniz, and Cantor-Kronecker. It is easy to portray Kronecker as the villain. To Mr. Aczel's credit he gives a fairly evenhanded account. However, it is probably true that the feud with Kronecker exacerbated Cantor's mental deterioration, though it is unlikely that it was the primary cause. To paraphrase Chesterton, "poets [and musicians] do not go mad; mathematicians and chess players do." Mr. Aczel discusses at some length if there might be truth to Chesterton's observation. It is easy to see Cantor as a kind of Prometheus who gave mankind knowledge of infinity, but was terribly punished by the gods.
The author, Amir D. Aczel, does not provide much information regarding his professional affiliations. The hints he does provide suggest he is a professional writer, with a strong interest in mathematics and theoretical physics. Not being a professional mathematician has its advantages when writing this kind of book. Professional mathematicians and scientists are very shy about transcending the borders of their discipline. Particularly "taboo" are religious or mystical speculations.
Mr. Aczel is free of such restrictions. This allows him to link the history of infinity with ancient religious/mystical thought found in the Kabbalah, St. Augustine and Dante.
In an age when ethnic origins, background and religious beliefs are supposed to be irrelevant to a person's professional accomplishments, there is an aspect to Mr. Aczel's book that might discomfit the "political-correctness hypersensitive." For example, it is possible to interpret his book as saying that Cantors familiarity with the Kabbalah allowed him to accept a completed infinity. Conversely, the Greeks, who gods all had limited powers, could not accept a completed infinity. Gutsy! He may be on to something, even though it opens the door a crack to some historically troubling concepts regarding the influence of religion and ethnicity on scientific thought.
When talented and intelligent writers embark on a new topic they often make egregious mistakes. This is to be understood - it is difficult to completely master a new topic involving deep concepts. Mr. Aczel's book is free of such mistakes (suggesting that he really does know his math). There are some minor ones though, apparently due to carelessness. For example, on page 20 he states, "irrational numbers have no patterns which repeat forever." This is not true. What he means is that irrational numbers cannot have endlessly repeating blocks of digits. However, they can have other simple repeating patterns, only slightly more complicated. On page 32 he states that there are ten permutations of the letters YHVH. There are twelve permutations.
All in all, a wonderfully stimulating book!
Interesting book, but not necessary
Aczel wrote this book in layman terms and he did a good job explaining the concept of transfinite numbers.
However, the association of the concept of infinity with another concept, God, is totally nonsense. (Same with the application of the Incompleteness Theorem to god.)
... Read more Isbn: 156858105X
Subjects: 1. 1845-1918
2. Biography / Autobiography
3. Cabala
4. Cantor, Georg,
5. Infinite
6. Logic
7. Mathematical Physics
8. Religious
9. Science
10. Science/Mathematics
11. Scientists - General
12. Set Theory
13. Spirituality - General
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$16.47
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A Great Read, and New Insights, Too
A prototype of bad popular science
