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Georg Cantor: His Mathematics and Philosophy of the Infinite
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.
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.
404 pages, Paperback
First published January 1, 1979
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Displaying 1 - 3 of 3 reviews
January 14, 2015
Hoo. Boy.
If, like me, you were someone who heard snippets of Cantor's life and work in various math classes or elsewhere, but wanted to learn a lot more, not only about who he was but in extreme detail about what work he actually did, this book is for you. But be warned to pay attention, because there's an awful lot of uncensored analysis (in the mathematical sense) whenever the author wants you to understand some part of Cantor's work. As such, if you didn't get relatively far through an undergraduate math program, you'll feel really lost. More so because naturally most of these results didn't make use of set theory, so things feel a lot harder to modern eyes than they have to be. If this isn't your speed, pick up instead.
Disclaimers aside, Dauben did a staggeringly impressive job at trying to get an accurate picture of Cantor in all facets of his life. You'll see Cantor not only as a disruptive mathematician, but also as philosopher, theologian, briefly as an artist, and with some more grounded insight into his many publicized breakdowns (spoiler warning: Dauben doesn't think that Kroenecker was a sufficient or even a necessary condition for these breakdowns). I've always thought that Cantor resembled Leibniz in a lot of ways, but at this point I'm more convinced that they're very nearly the same person. In Cantor you get a driven, explicitly religious motivation driving nearly all his work, and also helping him fend of the slings and arrows of finitists and intuitionists alike. This could also lead him astray, such as when he applied the full force of his indignation against infinitesimals, and produced a "proof" of their nonexistence which fell into exactly the same problem as a proof of Aristotle's he had rightly criticized regarding the existence of completed infinities. However, on the whole, Dauben makes the case that he was the perfect person to introduce set theory and the transfinite numbers to the mathematicians of his day, since it absolutely required the diligent work and strength of will Cantor possessed to get establish the existence of transfinite numbers around the protests of Kronecker and his followers and co-belligerents.
All in all, the book is a really in-depth look at Cantor's work (to the point where some people might call it pedantic at uncharitable moments when you might want to say "yes, we KNOW what alephs are, I don't need EVERY STEP of his first, probably unpolished work to get it!" [though it's certainly a nice middle ground between modern set theory and actually reading Cantor's Grundlagen. {} <- that's an empty set]). You'll get a really good look at the personality that led to set theory and the transfinite numbers, the actual work he did to get there, and even more illuminating, the mathematical and philosophical context Cantor was embedded in, and how the finitists, logicists, formalists, platonists, neo-Thomists and others burst into open conflict over something that now seems not just simple, but even "intuitive".
If, like me, you were someone who heard snippets of Cantor's life and work in various math classes or elsewhere, but wanted to learn a lot more, not only about who he was but in extreme detail about what work he actually did, this book is for you. But be warned to pay attention, because there's an awful lot of uncensored analysis (in the mathematical sense) whenever the author wants you to understand some part of Cantor's work. As such, if you didn't get relatively far through an undergraduate math program, you'll feel really lost. More so because naturally most of these results didn't make use of set theory, so things feel a lot harder to modern eyes than they have to be. If this isn't your speed, pick up instead.
Disclaimers aside, Dauben did a staggeringly impressive job at trying to get an accurate picture of Cantor in all facets of his life. You'll see Cantor not only as a disruptive mathematician, but also as philosopher, theologian, briefly as an artist, and with some more grounded insight into his many publicized breakdowns (spoiler warning: Dauben doesn't think that Kroenecker was a sufficient or even a necessary condition for these breakdowns). I've always thought that Cantor resembled Leibniz in a lot of ways, but at this point I'm more convinced that they're very nearly the same person. In Cantor you get a driven, explicitly religious motivation driving nearly all his work, and also helping him fend of the slings and arrows of finitists and intuitionists alike. This could also lead him astray, such as when he applied the full force of his indignation against infinitesimals, and produced a "proof" of their nonexistence which fell into exactly the same problem as a proof of Aristotle's he had rightly criticized regarding the existence of completed infinities. However, on the whole, Dauben makes the case that he was the perfect person to introduce set theory and the transfinite numbers to the mathematicians of his day, since it absolutely required the diligent work and strength of will Cantor possessed to get establish the existence of transfinite numbers around the protests of Kronecker and his followers and co-belligerents.
All in all, the book is a really in-depth look at Cantor's work (to the point where some people might call it pedantic at uncharitable moments when you might want to say "yes, we KNOW what alephs are, I don't need EVERY STEP of his first, probably unpolished work to get it!" [though it's certainly a nice middle ground between modern set theory and actually reading Cantor's Grundlagen. {} <- that's an empty set]). You'll get a really good look at the personality that led to set theory and the transfinite numbers, the actual work he did to get there, and even more illuminating, the mathematical and philosophical context Cantor was embedded in, and how the finitists, logicists, formalists, platonists, neo-Thomists and others burst into open conflict over something that now seems not just simple, but even "intuitive".
May 15, 2025
While a challenging read, this book gave me a fascinating insight into the process of mathematical investigation. Too often, ideas like Cantor's findings on infinite sets are presented as faits accomplis, but the truth is that they were shaped over decades of difficult work and acrimonious argument. Seeing into the life of someone while he was working these things out was well worth the time it took.
May 12, 2015
Wow. That is one dense book. Next step is to take a set theory course. Then re-read
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