Customer Reviews

The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (Dover Books on Mathematics)

5 star:		(2)
4 star:		(0)
3 star:		(2)
2 star:		(0)
1 star:		(0)

 

 

The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise by Mary Tiles is a fascinating mix of mathematics, mathematical logic, and philosophy that should appeal to (and challenge) both mathematics and philosophy majors at the undergraduate and graduate level.

The focus is on the Generalized Continuum Hypothesis (GCH); the reader will meet topics like numbering the continuum, developing Cantor's transfinite ordinal and cardinal numbers, evaluating the ZF axioms underlying set theory, and examining the work of Frege and Russell.

The first four chapters (The Finite Universe; Classes and Aristotelian Logic; Permutations, Combinations, and Infinite Cardinalities; and Numbering the Continuum) provide a historical, philosophical, and mathematical context for the more challenging chapters that follow. Some readers may wish to skip familiar sections although I found these early chapters to be quite engaging.

Chapter 5 - Cantor's Transfinite Paradise is a good, standalone introduction to Cantor's transfinite ordinal and cardinal numbers and to the General Continuum Hypothesis (GCH).

Chapter 6 - Axiomatic Set Theory is another good standalone chapter. Mary Tiles introduces the Zermelo-Fraenkel axioms that underlie modern set theory and develops a restatement of the GCH in the language of the ZF axioms.

Chapter 7 - Logical Objects and Logical Types delves deeply into the work of Frege and Russell. This was not the first time that I had encountered Russell's ramified type hierarchy, but nonetheless I still found this section slow going.

Chapter 8 - Independence Results and the Universe of Sets assumes substantial familiarity with model theory. Specific topics include Godel's constructible sets, cardinals and ordinals in models, inner models, and generic sets. Readers can either browse this technical chapter or omit it if they are willing to accept on trust the independence of the generalized continuum hypothesis and of the axiom of choice from the remaining Zermelo-Fraenkel set theory.

The final chapter, Mathematical Structure - Construct and Reality, summarizes the key philosophic issues underlying not only the generalized continuum hypothesis, but also with set theory in general and with the theory of transfinite numbers in particular.

I thoroughly enjoyed this introduction to Cantor's transfinite numbers. Mary Tiles has created an intriguing examination of the generalized continuum hypothesis.

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.

While this book is well-written and appears to provide a sufficient introduction to the concepts of set theory, it approaches but falls short of its titular subject- discussing the impact of set theory on philosophy.

While it addresses Cantor's project of putting analysis on a firm footing stripped of geometrical intuition, it neither addresses what additional benefit is to be gained in pursuit of this project, nor truly addresses the impact of set theory on philosophy. Instead, it makes an effective but somewhat pointless summation of aristotelian philosophical concerns, including also such things as Zeno's paradoxes, but does not address how formal set theory addresses these concerns other than drawing an entirely imaginary universe in which paradoxes are excluded by definition.

As such, set theory is relegated to the same realm as string theory- an interesting religion, entirely believable, but providing no testable benefit. Analysis and the concept of infintesimal limits and calculus provide concrete tools for challenging Zeno, even if they don't provide a definitive footing to formalize themselves.

This book would have benefited greatly from going full circle; returning to the world of philosophy in a post-Cantorian world. It ends, though, satisfied that it has explained set theory without making any attempt to tie it back to the earlier discussion of pre-Cantorian, pre-Russellian philosophy, as though Russell and logical positivism is the end of philosophy. Any book which has a title containing "philosophy" should require a chapter on "next steps", beyond simply suggesting additional reading, to give the reader a desire to pursue the topic further. Tiles' work fails on this account, despite being well written, and as such receives only a mediocre grade.

My interest in set theory and the foundations of Mathematics is already established around here, just look at my reviews, heheheh... The advantage of this text is the nice presentation of the logical foundations, even telling the history of concepts development. A nice reading for those who like the subject.