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The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise (Dover Books on Mathematics), by Mary Tiles
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A century ago, Georg Cantor demonstrated the possibility of a series of transfinite infinite numbers. His methods, unorthodox for the time, enabled him to derive theorems that established a mathematical reality for a hierarchy of infinities. Cantor's innovation was opposed, and ignored, by the establishment; years later, the value of his work was recognized and appreciated as a landmark in mathematical thought, forming the beginning of set theory and the foundation for most of contemporary mathematics.
As Cantor's sometime collaborator, David Hilbert, remarked, "No one will drive us from the paradise that Cantor has created." 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; logical objects and logical types; and independence results and the universe of sets. She concludes with views of the constructs and reality of mathematical structure.
Philosophers with only a basic grounding in mathematics, as well as mathematicians who have taken only an introductory course in philosophy, will find an abundance of intriguing topics in this text, which is appropriate for undergraduate-and graduate-level courses.
- Sales Rank: #362414 in Books
- Published on: 2004-06-18
- Released on: 2004-06-18
- Original language: English
- Number of items: 1
- Dimensions: 8.50" h x .58" w x 5.51" l, .62 pounds
- Binding: Paperback
- 272 pages
About the Author
Jim and Mary Tiles are authors of several books including "The Philosophy of Set Theory" and "Things that Happen,"
Most helpful customer reviews
43 of 45 people found the following review helpful.
Fascinating Introduction to the Generalized Continuum Hypothesis
By Michael Wischmeyer
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.
3 of 4 people found the following review helpful.
Wonderful
By James Squanderlast
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).
1 of 1 people found the following review helpful.
Philosophy AND mathematics
By Freethinker
Excellent book! Having some background in mathematics and philosophy will definitely help. I have been interested in these topics since the seventies and have never found a book as informative and interesting as this! I have found good philosophical books that were mathematically deficient and good mathematical books that were philosophically na�ve. This book is a rare find!
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