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

Rating:
Summary: Where's the beef ?
Review: A very wordy, but interesting book. The author intertwines the subject matter around a very well developed background, in fact a little too well developed. The main story is about the human mind's inability to contemplate infinity . The author points out that there are warnings in the Kabbalah, the Jewish book of mysticism, about peering into this aspect of mathematics. The famous mathematician Georg Cantor is credited with discovering and pioneering this area of mathematics. Mysteriously every time Cantor attempted to seriously delve into infinity theory he experienced a mental breakdown. Kurt Goebel another famous mathematician was also mentally affected by working in this area. The very basics of what they were considering was performing mathematical operations on magnitudes of infinities. Infinities of different sizes. It is a strange concept but something akin to asking, "what set has more members, a set of all the integers or the set of odd integers "? Once infinities are ordered then they may be ranked and considered eligible for mathematical operations. This is the second book I have read by Mr. Aczel. I believe the title of this book should have lead to the "meat" of the story, but the background history and the biography of Georg Cantor is given the same weight as the "mystery of the Aleph". The same was true of another book by Mr.Aczel, "God's Equation" the background material overshadowed the story. I would prefer a little more in-depth text about the heart of the matter.

Rating:
Summary: Aczel's books are disappointing
Review: Aczel has gotten a lot of press, so I read two of his books to evaluate whether to recommend them to my students. I do not recommend his work to anyone. His topics are interesting and important, but his treatment is very disappointing. Light and fluffy writing can help ease laypeople into reading science, if it is organized and accurate. Aczel's is not. He wanders among ideas without developing them in enough depth to appreciate. He poses interesting questions as if he intends to answer them, but does not.

Read Simon Singh or Timothy Ferris instead. Just as easy to read, and far more coherent and satisfying.

Rating:
Summary: Superb!
Review: Aczel writes in a way that is utterly profound, expounding on the mysteries of the maker of all infinity, the Blessed Ein Sof (the true essence of which can never be known to humans), at the same time delving into the mysteries of transfinite theory, and its creator, Georg Cantor, who went mad trying to concieve of that which can never be fully understood. Of all of the books that I have read, this is by far one of the most profound. You don't have to a Kabbalist or lover of mathematics to enjoy this book, Aczel writes in a way that makes the macrocosm of infinity easily understandable by the avearge reader. However, love of Kabbalah and math is a plus in understanding the profundity of the message the book conveys.

Rating:
Summary: Interesting book, but not necessary
Review: 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.)

Rating:
Summary: Complete Nonsense
Review: I would never have imagined that a quest this abstract could entail a story so human. Yet, that is exactly what Amir Aczel provides in this smooth tale of the many humbling encounters with the realm of infinity.

There are two lessons from this compact survey on the effort expended and the toll imposed on those bold enough to go where no person can go. First, the urge to comprehend infinity is an ancient quest and inextricably tied to the effort to ascertain the nature of God. Second, getting to know infinity can be massively bad for one's mental health. Mr. Aczel manages an almost impossible task (infinity tends to do that) in this text. He is (a) attempting to survey an enormous amount of the history mathematics and, to some extent, religion, and (b) providing a glimpse into the lives of those mathematicians that have ventured into this field. At the heart of this book is Georg Cantor, founder of modern set theory. Cantor sought to transcent an intuitive understanding of infinity. He sought an ordered system; specifically he sought to prove what became known as the continuim hypothesis: basically, that the lowest order of infinity (some cardinal numbere) was followed by the cardinal number, c (thus permitting Cantor to give ordere to his transfinite numbers). Against this hypothesis stood the possibility, urged by any number of Cantor's opponents, of infininty somewhere before one reached c. The search to prove what Godel later demonstrated to be an undecidable hypothesis may well have led Cantor (and Godel for that matter) to madness. At minimum it may have activated any underlying predisposition to mental illness in both men. They were not, as Aczels's discussion of the Kabbalists shows, inifinity's first victims. Aczel has provided a balanced and very human exploration into a topic that draws its victims as a moth to the flames.

Rating:
Summary: Trying to hard to ride the Kabbalah trend
Review: 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.)

Rating:
Summary: Aczel writes another winner
Review: 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.

Rating:
Summary: A delightful guide to the foothills of a huge subject
Review: This delightful little book is a Cook's Tour of some very important personalities in mathematics and their work on the concept of Infinity (actually various magnitudes of infinities, I guess), the Continuum Hypothesis, and the Axiom of Choice. While the author takes us back to the ancient Pythagoreans and their determination to keep irrational numbers secret knowledge, the story really centers around Georg Cantor and his struggles in founding the study of mathematics in this field. Cantor was a mystic as well and there is also more than one appearance of the Kabbalah.

Certainly, you can't learn the subject from this book. However, like visiting some vast architectural wonder that you can only take in as a big view, this book places lots of Post It notes on important points if you want to begin reading more deeply about these profound ideas. And if you don't, it is certainly a fund way to spend a few hours.

The author provides four pages of references for further reading, but if, like me, you don't know the field you will likely have to do preliminary studies to just get to the foothills of really taking on the subjects studied in this book. If you already understand the math then this book is likely too light for you unless you somehow missed out on the history of your field.

I enjoyed the book and if you are interested in how serious thinkers learned to think about Infinity and what it actually means, then this book is a fine initial guide.

Rating:
Summary: Disappointing
Review: While this book is at times entertaining, it did disappoint me. This is really too bad, since the author is well versed in the content matter of this book.

The books picks up some players in mankind's quest to come to grips with the concept of infinity. It focuses on the ancient Greeks, Jewish mysticism and modern mathematics. To juice the narrative up, the author chose to portray two of his characters, Cantor and Goedel as the real live counterparts of Adrian Leverkuhn in Mann's Doctor Faustus.

It is obvious that some considered this sloppily edited book a success. It may be that mathematicians who are fully versed in the mathematical subject matter, were entertained by the cute inclusion of Kaballah and small biographies. Unfortunately, I have only the basic knowledge of undergraduate coursework in real analysis and set theory. Based on this and a similar level of knowledge of Jewish mysticism, I can not help but consider this book an utter failure.

The history of science has been one of disagreement and often feud. The same is true regarding the development of a great many of the cornerstones of current mathematics. The line of reasoning followed here regarding the fate of Cantor and Goedel is on the same level as that of recent publications on the already dispelled magic of "Bible codes".

In addition, the clear but casual treatment of many of the important mathematical milestones in the book, simply does not give credit to and insight into the struggles that these mathematicians went through to find order into what previously appeared to be chaos.

Well too bad. Let's hope that Simon Singh will one day revisit the subject of this book.

Rating:
Summary: Good historical reasearch combined with original thinking
Review: 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!