Rating: 
Summary: Defies Intuition
Review: 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.
Rating: 
Summary: excellent discussion of infinity in mathematics
Review: The author does a very good job at showing how Cantor was intrigued by the infinite. He gives a chapter on Kabbalah to give the reader information on the concept of infinity. He does this to give the reader background on how awesome such a concept is. The book tells how Cantor went insane trying to understand the infinite. He was hospitalized with depression many times when he came to close to the infinite. What intrigued me most about the book is Cantors mathematics. Cantor was trying to show by this discipline what it means to be infinite. He writes about lower and higher sets of infinity. To make the reader understand what this means, we turn to philosophy. What does it mean for instance, when we say that God has infinite knowledge? Most people would probably say that God has knowledge of endless things. This is true. In a deeper sense though it means that God has infinite knowledge of infinite things. In other words, if we took a subject like mathematics, God would know infinite knowledge of algebra, infinite knowledge of calculus, and so on. Such knowlege truly boggles the mind. Cantor went insane because he was trying to go above his head. He was trying to limit the infinite to his understanding, and this is impossible. What is limited can never take in the infinite. All we know is that God cannot make a contradiction, but trying to understand the infinite can never be done. All we can do is take in small doses at a time. This book is one math book I highly recommend. It makes for the most intresting reading. It is far more fascinating then any fiction.
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: An excellent combination of math, psychology, and Cabalah
Review: 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.
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!
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