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The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise

Mary Tiles
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Dual Review:
Howard Delong, A Profile of Mathematical Logic, Dover 1970/2004
Mary Tiles, The Philosophy of Set Theory, Dover 1989/2004

Dover, that much-beloved re-publisher of books from out of the past in downright cheap editions, has just reissued two books on logic and set theory which have a lot to recommend them even as their objectives are rather different. Howard Delong's A Profile of Mathematical Logic is a serious, complete text suitable for a second course in logic. It's so good that it would be hard to find a better book for such a course, despite the existence of many competitors. And Mary Tiles' book, The Philosophy of Set Theory, subtitled "An historical introduction to Cantor's paradise," might qualify as a nice prelude to Delong's text, but it is more than that: it is a beautifully written, ambitious work which succeeds in its undertaking to explore Cantor's work from different perspectives and with a true philosophical flair. It is never superficial even as it is accessible (modulo some effort on the reader's part); this fits with the author's goal of reaching undergraduates in mathematics and in philosophy. Indeed Tiles manages to be encyclopedic without being pedantic, even given that her intended audience is the usually fuzzy set of undergraduates in mathematics or philosophy. More about this presently.

First, Delong's book. As I indicated above, the treatment of logic as such is marvellous. Metamathematics is covered beautifully indeed (see Delong's essay on Church's thesis), and philosophical matters receive serious, if idiosyncratic coverage. It is in fact the author's secondary desire to engage philosophers in polemics, as it were, given their notorious occasional (or frequent?) tendency to relegate mathematical logic to a place of philosophical irrelevancy. Delong wages spirited war against this kind of imperialism and I'm happy to report that at the end of his book even the existentialists take a beating.

Tiles, also, cares what philosophers think, or what they should think, but she seeks to instruct rather than to correct (or worse). Although she is not particularly polemical she does work with counterpoint. In fact, her very title leads one to ask right off if the phrase used is actually well-defined: are we dealing with philosophy per se or with set theory qua mathematics? How would a philosopher regard set theory (evidently in ways that chafe Delong)? Conversely, does a mathematician really need to worry about any philosophical aspects of set theory?

Mary Tiles, writing some fifteen years ago from the University of Hawaii, a locale that would make one wax philosophical about paradise, certainly holds that these types of questions are of some importance; I agree wholeheartedly (as an avowed Platonist). However, unlike Delong, who pulls no punches, Tiles seeks to instruct, rather, and enlighten. I think that both approaches are valid, pretty much as a function of the philosopher at the receiving end of this Socratic dialogue. Nonetheless, Tiles does not eshew controversy, nor does she beat around the bush. Her topics include whether set theory (or mathematics) is created or discovered, the perspective of finitism, the "Absolutely Infinite," and of course such standards as Zermelo-Fraenkel set theory, Frege's approach (and Russell's), and Gödel's work. But her discussion always revolves around a (terrific) treatment of Cantor's work properly so-called (as the philosophers say).

I am happy to recommend both of these books.

Michael Berg teaches at Loyola Marymount University in Los Angeles.

1.The Finite Universe
2.Classes and Aristotelian Logic
3.Permutations, Combinations and Infinite Cardinalities
4.Numbering the Continuum
5.Cantor's Transfinite Paradise
6.Axiomatic Set Theory
7.Logical Objects and Logical Types
8.Independence Results and the Universe of Sets
9.Mathematical Structure--Construct and Reality
Further Reading
Glossary of Symbols