26 of 26 people found the following review helpful:
5.0 out of 5 stars
Great ... and a few warnings, December 8, 2006
This review is from: Georg Cantor (Paperback)
This is the definitive book about George Cantor, the brilliant mathematician whose work includes the groundbreaking development of both set theory and transfinite numbers.
Interestingly, the author's preface says this is not a biography of Cantor, though it does include personal information, especially as it relates to Cantor's intellectual development and emotional issues. Rather, it's a thorough and rigorous exposition of his mathematical and philosopical ideas. Dauben says, "... this book represents a study of the pulse, metabolism, even in part the psychodynamics of an intellectual process: the emergence of a new mathematical theory".
But, a few warnings. While both the Amazon and jacket blurb claims this is for the "general reader", it is not. It is most definitely NOT a popularization, and I don't think the publisher tries to make that clear. It is a scholarly tract, an extension of Dauben's Harvard doctoral dissertation, and it seems he has not watered it down much. It is highly technical, with many equations, and is primarily written for academicians who are fluent in higher mathematics (clearly, not a large potential audience for the book!). Consistent with such a scholarly publication, it includes excellent index, bibliography, and notes sections, with many entries being technical, from obscure journals, and/or in foreign languages.
I found that my three semesters of college calculus (though no set theory) were inadequate preparation to follow many of the mathematical arguments. If you have an undergraduate or higher degree in pure mathematics, you should have no trouble.
Dauben also uses a fair amount of German, and a little French and Latin, all without translation -- you're expected to know these things.
It's possible to get a sense of Cantor's accomplishments by simply skipping over the math and foreign languages that are beyond you, although the more prepared you are in these areas, the more you'll get out of the book.
However, if you're interested in the history of math but want to avoid the naked technicalities, I instead recommend William Dunham's "Journey Through Genius", which uses nothing beyond high-school mathematics. Dunham's book has twelve readable chapters on significant mathematical discoveries, and as a measure of Cantor's importance, he, like Euclid and Euler, gets two chapters while Archimedes, Newton, and the rest get just one.
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12 of 12 people found the following review helpful:
4.0 out of 5 stars
Hard core and worth it, March 11, 2004
This review is from: Georg Cantor (Paperback)
I found this was an excellent memoir of Cantor and his ideas. It goes far more deeply into the mathematics than other discussions of Cantor, and makes you want to read more about both set theory and about the development of topology in the early twentieth century. It also attempts to place Cantor's philosophy and mathematical ideas into a psychological context as well, which is probably appropriate - even essential - in the case of Cantor. Though, in twenty years, this aspect of the book may not wear well. All in all, it is a refreshingly strong and insightful treatment of one of the major historical figures in nineteenth century mathematics.
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9 of 9 people found the following review helpful:
4.0 out of 5 stars
Philosophy that can be mathematically proven, April 29, 2004
This review is from: Georg Cantor (Paperback)
Some areas of mathematics can be described as philosophy that can be proven, and that certainly applies to the transfinite arithmetic created by Georg Cantor. When his results were first published, they were met with a great deal of opposition by many established and influential mathematicians. Considering the revolutionary aspects of his work, the fact that there was opposition was not surprising and indeed necessary for mathematics to properly advance. However, the opposition to Cantor's main results was so strong that the story that it drove him mad has become a part of the mathematical folklore. As Dauben is careful to point out, his research indicates that Cantor would have suffered from the bouts of depression independent of the degree of opposition to his work and most likely independent of what kind of work he did. In fact, Cantor himself is quoted as stating that his time spent in institutions was some of the most restful and productive of his life.
Cantor was in fact a very strong personality, he stood up well against the opposition and ultimately was proven to be correct. Dauben not only explains the sources of that strength, from his supportive, yet not domineering father to his wife and children. His religious beliefs were also very strong, to the point where he firmly believed that his work was part of God's structure of the universe and he was nothing more than a messenger.
However, it is the mathematics that really matters and Dauben does an excellent job in describing the essence of Cantor's discoveries. These are complex topics, and yet he does a good job in explaining the fundamentals of the mathematics and why it was so difficult for other mathematicians to accept. This is the best description of Cantor's work at a general level that I have ever read.
Published in Journal of Recreational Mathematics, reprinted with permission.
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