When Newton and his successors defined the calculus in the 17th and 18th centuries, they were quite cavalier about infinities. For example, they treated sums of infinitely many numbers essentially the same way they treated sums of finitely many numbers. And when talking about derivatives, they were content to talk of changes over infinitely shrinking intervals without quite saying what they meant by that. As mathematics developed increasingly abstract methods, more divorced from the simple observations of physics, many problems cropped up, mostly having to do with the careless use of infinities. In order to deal with these problems, mathematicians devised precise definitions which made no explicit use of infinities.
But the new methods made it necessary for mathematicians to consider the sets of points where the methods broke down. In investigating them, Cantor had to consider infinite sets and even had to compare different sizes of infinity. Understandably, many mathematicians were upset. But others found Cantor's mathematics useful and worked to put set theory on a solid basis. A new theory, the Zermelo-Frankel theory of sets, was the result. It's not perfect, but it's good enough for most mathematicians. Most mathematicians today are quite comfortable with infinite sets.
I mention all this because Tiles doesn't. Given the subtitle, "An Historical Introduction to Cantor's Paradise", I was expecting to read about the intellectual climate in which Cantor developed his theory. So I am writing this book for the sake of anyone who might like a book putting Cantor's theory into its historical context: this is not the book.
It is a book about the philosophy of finitism, from ancient Greek times to the 20th century. It might be a good book on that topic; I have no way of knowing. For that reason, I can't fairly rate the book and the three stars shouldn't be taken seriously.
(My degrees are in math, not philosophy; for more about that, click on my name at the head of this review. Math texts are strong on theory but, unfortunately, weak on context.)
(Original review 25 Nov 2005; this paragraph 11 Jan 2006) The book makes several references to Zeno's paradoxes. These are based on an assumptiion which is incorrect in a universe, such as ours, where quantum effects are fundamental. Hence they are of purely philosophical interest.