The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity

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.

As a professional mathematician I can tell you that Aczel has done an admirable job. He tackles in this book some of the thorniest issues in modern mathematics: the axiom of choice and the continuum hypotheis. Aczel's explanations of topics such as Russell's paradox, Zermello's set theory, and Godel's work are among the best I have seen. He tells the story of the continuum hypothesis with verve and style. I strongly recommend this book to anyone who is interested in the history of mathematics.

This is a very enjoyable book about infinity and its history. It reviews the genesis of the concept of infinity from early times (ancient Greece), through the Middle Ages and on to the modern era. The author carefully surveys the major developments in this history, including the works of Galileo, Bolzano, Cantor, Godel, and others. He adds interesting asides about number mysticism both in the Jewish Kabbalah and in Christian teachings. The mathematics is kept at a low, understandable level, and the biographies of the major figures in this 2,500 year old drama of human understanding are very well developed. Overall, this is a readable and informative book. My only wish as a reader would have been to have more space devoted to the psychology of the actors in this drama. Many of these mathematicians went crazy. The question in every reader's mind is: Why?

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.

People have tried this for a few thousand years to understand the infinite, most along religious lines. _The Mystery of the Aleph_ (Four Walls Eight Windows) by Amir D. Aczel traces the history of these understandings, but concentrates on the mathematical understanding that was really begun only in the last century. Galileo contemplated two sets, the counting numbers 1, 2, 3, 4... and the square numbers 1, 4, 9, 16.... He found that every square from the second set could be paired with a number from the first: 1/1, 2/4, 3/9, 4/16, and so on. This means that although there is an infinity of numbers in either set, one set is exactly as big as the other. Galileo was shocked that this was true, even though it seems as if there are many more numbers in the first set; but he had found the key property of an infinite set, that it can be equal to a set included within itself. Bernhard Bolzano built on this strange finding to show that a line one inch long has as many points as a line two inches (or any number of inches) long. Georg Cantor is the mathematician most identified with studying infinities. Aczel's book is pretty good at explaining his very peculiar findings. Cantor found, for instance, that the infinity of counting numbers could be placed in a one to one correspondence with fractions (rational numbers). Of course, the fractions are more dense, given all of them that exist between only, say, 1 and 2. But the number of such fractions does not exceed the number of counting numbers. Cantor also had clever demonstrations that a one inch line had just as many infinite points on it as a one inch square plane, as did any size line and any size plane; the same was true of higher dimensions as well. This would seem to indicate that all infinities are the same size; however, Cantor showed that this was not true. Specifically, he showed that although the rational numbers could be paired up with the counting numbers, there were not sufficient pairs to be made if you included such numbers as the transcendental irrationals pi or e. Cantor went mad, and died in a psychiatric hospital; it is too much to say that contemplating infinities made him crazy, but his continued attempts to prove his Continuum Hypothesis provided increasing frustration, as did attacks from his fellow mathematicians. Gödel himself showed that the continuum hypothesis could not be proved or disproved. During his work on this problem of infinities, he began to go mad as well, showing his own symptoms of paranoia and obsessiveness. Eventually, he was convinced that his food was poisoned and he would touch less and less of it; he simply starved himself to death.So open up these pages if you dare; studying infinities has not been healthful for everyone. Aczel, however, does not go deeply into proofs, using good illustrations to provide access to non-mathematicians for some distinctly strange mathematical ideas.