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Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics), by Paul Taylor
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Practical Foundations of Mathematics explains the basis of mathematical reasoning both in pure mathematics itself (algebra and topology in particular) and in computer science. In addition to the formal logic, this volume examines the relationship between computer languages and "plain English" mathematical proofs. The book introduces the reader to discrete mathematics, reasoning, and categorical logic. It offers a new approach to term algebras, induction and recursion and proves in detail the equivalence of types and categories. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries across universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.
- Sales Rank: #2770843 in Books
- Published on: 1999-05-13
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x 1.46" w x 5.98" l, 2.24 pounds
- Binding: Hardcover
- 588 pages
Review
"Taylor paints rhapsodically on a broad, richly detailed canvas replete with examples and exercises. He invites readers to dip in at any point and structures his book accordingly. He embroiders his text with a running commentary that often fascinates...this book covers important ground in an original style." Choice
Most helpful customer reviews
12 of 14 people found the following review helpful.
Useful summary, poor exposition
By J. Elliott
The author's staccato writing style is reminiscent of Gilbert Strang's. Some may like it, but I find it jarring. The content is a concise summary of interesting topics at the confluence of mathematics, logic and computer science (see the table of contents), but it reads like a précis for those who already know the subject. This is no doubt fine if you fall into that category. If you're looking for an expository text, this, alas, isn't it.
16 of 20 people found the following review helpful.
A confusing hodgepodge of tangential ideas
By jasonc65
I agree completely with J. Elliott. The author states so many propositions without proof, and even the proofs given are too sketchy, forcing the reader to fill in every detail, and in many instances, the author's proofs are simply wrong. Many of his definitions are vague and confusing, in many cases bewildering the reader's mind with all kinds of tangential questions unrelated to the main topic. Paul Taylor misleads the reader with chapter titles like "Posets and Lattices" and "Cartesian Closed Categories" in which he does not stick to the topics he promises to cover but jumps all over the place into unrelated fields. It's like he wants to "introduce" the reader to so much that he has no time to explain anything.
Besides, there are so many better books for any of the subjects the book brings up. For category theory, there is "Categories for the Working Mathematician" by MacLane; for lambda calculus, there is Barendregt's, for topos theory, there is "Topoi" by Goldblatt, who does not prove everything he states, including several fundamental theorems, but at least he stays on topic; or if one simply wishes to forget about new approaches to foundations and take up traditional set theory, there is Jech, whose book is very difficult, but at least it it challenging. But as for Taylor, his is neither interesting, nor enlightening, nor even challenging. As for those who already "know it all", what's the point?
In short, the author does not start with the basics and build up in any sort of cumulative fashion, but diverts the reader's interests into every specialization into which mathematics is expanding. "A practical foundation of mathematics" is anything but foundational. Tempus est legendi aliud.
3 of 3 people found the following review helpful.
An Enjoyable and Insightful Exposition
By Hans-Martin Will
This is a rare kind of book found in current mathematical writing - a book that motivates, seeks connections across disciplines, and attempts to paint the broader picture of how the individual concepts and results are the pieces of a much larger puzzle. It is obvious that the author is on a mission, namely to convince the reader that contemporary mathematical foundations are important, applicable and useful to formulate, investigate and answer questions in mathematics and computer science. The book does so by drawing on many examples picked from widely across those disciplines, not only in the main text but even more so in the exercises.
Particularly in the light of more critical reviews of this text, though, I must admit that this is not the first book I have been reading on the material covered here. But it is surely one of the more enjoyable and insightful expositions in this field.
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