Book Description

One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula.

Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.

Customer Reviews (4)

Great ... and a few warnings
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.

Cantor,A Top Mathematician Who Unifies God and Sciene
This is a great book on the work of Georg Cantor.

The new millennium brought forth a new human hope -- to unify `that which is physical' with `that which is spiritual.' This hope would unify the fields of science and theology. Currently, these fields are separated by their different beliefs and methods of truth seeking. On beliefs, many scientists do not believe in God whereas all theologians believe in God. On truths, scientists use the scientific method to express true scientific statements whereas theologians use feelings, phenomena, and scriptures to express true theological statements.

I believe that all `big discoveries' unify 'that which is physical' with 'that which is spiritual.'For instance, the big discovery of Nicholas of Cusa unifies an infinite God and our finite universe. And, the big discovery of Leibniz unifies spiritual atoms with all physical things in the universe. Further, the big 1920 discovery of symbolic languages allows us to speak about God and the universe using the same symbolic language. In this book on Cantor, one earns that the mathematics and infinities of Georg Cantor is another `big discovery' that unifies God and the universe.

Cantor's theory of the infinity reveals two infinities. One is the `non-genuine' infinite. This infinity is incomplete. It applies to variable magnitudes in the universe, as they either grow beyond all limits or diminishes to an arbitrary smallness. This infinity always remains finite. Cantor's transfinite numbers orders and counts all non-genuine infinities. On the other hand, the `genuine' infinity is determinate, unchanging, and complete. It is found in the complex variables used by scientists. There, the determinate infinity is the single point at infinity. Complex variables in the neighborhood of this single point are thus analytical. The genuine infinity is also used by theologians to represent God.

It is obvious that organized research on these big discoveries must be increased considerably by all nations. To start organizing this research effort, I am writing a book on symbolic languages and the big discoveries of Cusa, Leibniz, and Cantor.

Philosophy that can be mathematically proven
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 andDauben 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.

Hard core and worth it
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 notwear well. All in all, it is a refreshingly strong and insightful treatment of one of the major historical figures in nineteenth century mathematics. ... Read more

Book Description

Customer Reviews (2)

Detailed Axiomatic Development of Transfinite Numbers - Not Suitable as Introduction
Georg Cantor's final and logically purified memoir on transfinite numbers was published in the late 1890s.This Dover reprint is the 1915 English translation by the mathematician Philip E. B. Jourdain; it also includes a lengthy, technically diffcult introduction by Jourdain.

Contributions to the Founding of the Theory of Transfinite Numbers is not suitable as an introduction. I unwisely disregarded caution from an earlier reviewer that Cantor's work would not be appropriate for a beginner in set theory. (I thought that I was reasonably acquainted with set theory, but I do admit that I was not a math major.)

The 82-page introduction by Jourdain assumes that the reader is reasonably familiar with the work of key nineteenth century mathematicians.While it is possible to skip the introduction, Jourdain's context setting is quite helpful. Cantor's transfinite numbers are so innovative and so unexpected that it almost seems as though they spring forth in a vacuum, but Jourdain shows that the earlier work of Dirichlet, Cauchy, Riemann, and Weierstrass helped point the way for Cantor.

Cantor's memoir (that is, his two-part discussion of transfinite number theory) comprise the remaining 125 pages. The difficulty with Cantor's axiomatic presentation is two-fold. First, the material itself is not easy - despite Cantor's careful approach. I even bogged down for awhile on his early discussion of the exponentiation of powers and how this leads to aleph-zero. And second, much of his terminology is outdated and unfamiliar. For example, there is no mention of sets, just aggregates and parts. Another example is that Cantor speaks of reciprocal and univocal correspondence. I have yet to complete Cantor's work, but I am continuing to plod along.

A recommendation: A much better starting point for readers new to transfinite numbers is a fascinating book by Mary Tiles, titled The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise. This work targets mathematics and philosophy majors, but is accessible to others.

An interesting book
There's nothing like reading the original.Here is the abstract theory of transfinite ordinals described by its originator, Georg Cantor.

It's probably not the best introduction to set theory for a beginner.The bookfocuses more on ordinal numbers than on cardinals or general sets.It'snot a great reference, either, since so many important results in settheory have been proven in the 100 years since Cantor.But I like thisbook a lot nonetheless.The exposition is beautiful -- concise, clear, andlogical.It's one of the most nicely presented math books I've read. ... Read more

Book Description
Das Buch behandelt die Basis-Resultate der Mengenlehre aus der Zeit Cantors und Zermelos, was etwa den Zeitraum von 1870 - 1930 abdeckt. Die Ideen dieser Zeit bilden das Herz der Disziplin und haben das heutige Bild der Mathematik entscheidend mit geprägt.
Ziel ist, die zentralen Konzepte und Probleme der Mengenlehre - Mächtigkeiten, Kardinalzahlen, Kontinuumsproblem, Wohlordnungen, transfinite Zahlen und transfinite Rekursion, mengentheoretische Untersuchungen von R - in ihrem Wesen begreifbar zu machen. Eine Axiomatik wird in Übereinstimmung mit der historischen Entwicklung erst dann eingeführt, wenn die Theorie bereits weit gediehen ist und nach einem stabilen Fundament verlangt. Schließlich wird die Axiomatik in einen formalen Rahmen eingebettet, was Resultate über die Grenzen des Gebäudes ermöglicht (wie z.B. die Unabhängigkeit der Kontinuumshypothese). Das Buch wendet sich an Studenten (Lehramt und Diplom) und Dozenten der Mathematik.
Jetzt in verbesserter und erweiterter Auflage. ... Read more