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Editorial Review Product Description
A compelling narrative that blends a story of infinity with the tagic tale of a tormented and brilliant mathematician. From the end of the ninteenth century until his death, one of history's greatest mathematicians languished in an asylum, driven mad by an almost Faustian thirst for universal knowledge. THE MYSTERY OF THE ALEPH tells the story of Georg Cantor (1845-1918), a Russian born German whose work on the 'continuum problem' would bring us closer than any mathemetician before him in helping us to comprehend the nature of infinity. A respected mathematician himself, Dr. Aczel follows Cantor's life and traces the roots of his enigmatic theories. From the Pythagoreans, the Greek cult of mathematics, to the mystical Jewish numerology found in the Kabbalah, THE MYSTERY OF THE ALEPH follows the search for an answer that may never truly be trusted.Amazon.com Review
The search for infinity, that sublime and barelycomprehensible mystery, has exercised both mathematicians andtheologians over many generations. Jewish mystics, in particular,labored with elaborate numerological schema to imagine the purenothingness of infinity, while scientists such as Galileo, the greatastronomer, and Georg Cantor, the inventor of modern set theory (aswell as a gifted Shakespearean scholar), brought their training tobear on the unimaginable infinitude of numbers and of space, seekingthe key to the universe.In this sometimes technical but always accessible narrative, AmirAczel, author of the spirited study Fermat's Last Theorem,contemplates such matters as the Greek philosopher Zeno's severalparadoxes; the curious careers of defrocked priests, (literal) madscientists, and sober scholars whose work helped untangle some ofthose paradoxes; and the conundrums that modern mathematics hassubstituted for the puzzles of yore. To negotiate some of thoseenigmas requires a belief not unlike faith, Aczel hints, noting, "Wemay find it hard to believe that an elegant and seemingly very simplesystem of numbers and operations such as addition andmultiplication--elements so intuitive that children learn them inschool--should be fraught with holes and logical hurdles." Hard tobelieve, indeed. Aczel's book makes for a fine and fun exercise inbrain-stretching, while providing a learned survey of the regionswhere science and religion meet. --Gregory McNamee ... Read more Customer Reviews (56) 
Religious and Secular Conceptions of the Infinite
The book went way over my head.I read it, and have to admit that I didn't understand much of it.I rated this one a five because I stand in admiration of smart people like Dr. Aczel in the world, who think about, ponder and write books about subjects like the mathematical understanding of infinity.I'm lucky if my checkbook balances right every week.
Very Good book about the Infinity and the Continuum Hypothesis
I enjoyed reading the book very much. In general, any book about numbers, and its development in human history amazes me. The story of Mr. Aczel is not an exception. The struggle of Georg Cantor is clearly elaborated in details in this book. How he goes mad to prove the cardinality of the real numbers. How many are there really? The book also makes a good job in demonstrating the early studies of Jewish mystics to explore the infinity and God. Yet one thing that looked weak to me is the connection between infinite numbers, God and Kabbalah. I wish the author used more examples to give the message that he is trying to make. I am personally very interested in Sufism (a sister trend or movement) so for me reading the book was definitely informative and inspiring to a certain degree. Yet I expect similar books from Mr. Aczel and especially a follow-up sequel to this book to make things clear.
early maths theory
Very interesting history of the development of mathematical ideas,especially the existence of irrational numbers,and the idea infinity can be approached and used but never reached.....
mathematics, cantor and mysticism
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.
Well-written and interesting, but somewhat superficial
First, the good news. Aczel's book -- part biography, part history of infinity, part primer of some of the more challenging concepts in mathematics -- is engaging and well written. Much better written, in fact, than many similar books on the history of or on topics in mathematics that I've read. He has a lively style that keeps you turning the pages, and he is generally very good at simplifying complex axioms and proofs for the layperson. The short précis of the concepts of infinity among the ancient Greeks and Jews is pretty captivating subject matter, too. And the short biographies of the key mathematicians chasing the infinite are all sound and worthwhile.
Now, the bad news. Considering that the subtitle of the book invokes the Kabbalah, Aczel gives it rather short shrift. He endeavors to summarize the subject, particularly in relation to things infinite, but does so too carelessly. I wanted more elaboration on that. Then he attempts to bring the Kabbalah back from time to time, as with Cantor's debatably Jewish heritage and with the diaspora of the Jews during World War II, but these connections are only hinted at. They feel superficial and without the persuasive weight to justify their inclusion. Also, I feel Aczel is a bit too baldly assertive in blaming Cantor and Gödel's mental problems on their struggles with the Continuum Hypothesis. Might it not have been the other way around, latent mental instability leading these two men to that particular compulsive struggle? I understand that pointing a finger at Infinity and shouting "j'accuse!" makes for more dramatic nonfiction, but it comes at a cost in accracy, doesn't it?
Still, despite these complaints, I can marginally recommend the book as an interesting read on the history of the notion of infinity. Or at least parts of that history.
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