Mary Tiles

Wow. This is excellent. Tiles traces the history of the concepts of infinity and infinitesimals from Xeno's paradoxes through "pathological" non-continuous analytic functions and into Cantor's work on transfinites. The book is framed as a mathematically-informed philosophical work, and this is most true of the first few chapters which do an excellent job of elucidating the differences between "potential" and "actual" infinities, and how these inform the debate over whether mathematics is invented or discovered. As the book moves on to Cantor's work, it becomes more of a strictly mathematical exposition of alephs and omegas. This is not a criticism though: the chapter on the Zermelo-Frankel axioms and how they can be used to build Cantor's hierarchy of ordinals is the clearest treatment of this material I've ever seen.

The exposition is clear but brisk, and though Tiles defines all her terms this book might be rough going to someone who has never encountered these ideas before. However, for a reader with a second-hand knowledge of how set theory, logic, and infinity are entangled, The Philosophy of Set Theory is an excellent way to tie all the loose ends together.

Errata:
Chapter 5, Section 3, page 108, paragraph 1
"...whereas the ordered union B+A..." should read "...B+A = ..."

Chapter 6, Section 2, page 122
The expression for the Sum Set Axiom looks incorrect to me. There is a for all x quantifier, but no x variable in the expression, which contains an unbound a variable which I think is supposed to be an x.… (plus d'informations)