Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics) [Hardcover]

Paul Taylor (Author)

Book Description

ISBN-10: 0521631076 | ISBN-13: 978-0521631075 | Publication Date: May 13, 1999

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.

Editorial Reviews

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

Book Description

This book is about the basis of mathematical reasoning both in pure mathematics itself (particularly algebra and topology) and in computer science (how and what it means to prove correctness of programs). It contains original material and original developments of standard material, so it is also for professional researchers, but as it deliberately transcends disciplinary boundaries and challenges many established attitudes to the foundations of mathematics, the reader is expected to be open minded about these things.

Product Details

• Hardcover: 588 pages
• Publisher: Cambridge University Press (May 13, 1999)
• Language: English
• ISBN-10: 0521631076
• ISBN-13: 978-0521631075
• Product Dimensions: 9.4 x 5.9 x 1.7 inches
• Shipping Weight: 2.2 pounds (View shipping rates and policies)
• Average Customer Review: 2.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #2,396,659 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

11 of 12 people found the following review helpful:

2.0 out of 5 stars Useful summary, poor exposition, March 26, 2006

By 

J. Elliott (Natick, MA USA) - See all my reviews
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This review is from: Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics) (Hardcover)

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.

12 of 14 people found the following review helpful:

1.0 out of 5 stars A confusing hodgepodge of tangential ideas, August 9, 2006

By 

jasonc65 "[e-mail]" (Wilmington, DE USA) - See all my reviews

This review is from: Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics) (Hardcover)

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.

2 of 7 people found the following review helpful:

5.0 out of 5 stars A Superlative Compendium of Essentials in Logic and Th. CompSci, August 27, 2008

By 

Farogh Dovlatashahi "True Enquirer" (Los Angeles, USA) - See all my reviews
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This review is from: Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics) (Hardcover)

This is a superlative book, a compendium of absolutely essential topics in the range between mathematics,philosophy, logic and theoretical computer science. This is a complex field. We often find writers addressing philosophical, logical and practical issues relating to logic, its implementation, the relation bt. theory and "reality" of reasoning, mathematical aspects at the hight end of pure mathematics: sheaves, topology, algebraic logic, ... and so on and on.

To those who do not enjoy working in areas of enquiry that have multiple, perhaps indefinitely multiple aspects to them, there are welcome to write a book that unifies it all!