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The Philosophy of Set Theory: An (Dover Books on Mathematics) [Paperback]

Mary Tiles

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Book Description

24 Sep 2004 0486435202 978-0486435206 Dover Edition
This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory, and more. 1989 edition. Includes 32 figures.

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Most Helpful Customer Reviews on (beta) 4.3 out of 5 stars  6 reviews
37 of 39 people found the following review helpful
5.0 out of 5 stars Fascinating Introduction to the Generalized Continuum Hypothesis 24 Dec 2005
By Michael Wischmeyer - Published on
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.
46 of 63 people found the following review helpful
3.0 out of 5 stars Finitistic Philosophy, with Some Comments on Cantor 25 Nov 2005
By Edward F. Strasser - Published on
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.
21 of 30 people found the following review helpful
3.0 out of 5 stars good- but not true to title 22 Aug 2006
By Matthew E. Harbowy - Published on
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.
1 of 1 people found the following review helpful
5.0 out of 5 stars Wonderful 2 Oct 2013
By James Squanderlast - Published on
Format:Kindle Edition
I'm a math enthusiast with no formal training in math. I've only studied finance/econ. So we use some areas of math heavily, but don't tend to play around with the more pure analysis of math. I was enamored by set theory in the past, but the books I bought were just too intense. I lacked not only the background, but also the time to teach myself the intense complexity of high level set theory. However, I absolutely loved the insight even basic set-theory analysis offered me into calculus, which I gained from the back four pages of a game theory textbook I owned!

This book was wonderful. It was conceptually challenging, and not uncommon for me to spend 5 minutes on a page. At the same time, it's the type of book a clever person without an intense math background could take to a coffee shop. I also found the historical philosophy parts of the book fascinating, and offering a wonderful foundation for the reasons new models and ideas were formed.

I initially found this book at a bookstore, and decided not to buy it, as I made the pretentious observation the author/professor did not teach at a 'prestigious school.' I then doubled back and made the impulse purchase after getting hooked on one of the chapters in the middle of the book, and decided to give it a proper shot. I have to say, I absolutely love this book. The insights into critical analysis of algebra/geometry/infinite series, has legitimately helped me in my work in game theory. While I hope to eventually study a high level course in real-analysis, this book manages to be both captivating and as rigorous as possible without creating a full-blown mathematical textbook.

I highly recommend it to anyone who loves set theory OR philosophy, and wants to self-study more (PS: If you're a philosopher with no math background, I still suggest you buy this book. The best way for philosopher to learn math is to start in areas that overlap with philosophy as opposed to being discouraged from college courses that focus on boring exam based computation rather than critical analysis).
5.0 out of 5 stars He really liked it. 14 July 2014
By Beth Swansboro - Published on
Format:Paperback|Verified Purchase
It was for my husband. He really liked it.
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