Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between 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.
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Review of the hardback: 'This is a fascinating and rewarding book … each chapter has several pages of subtle, provocative and imaginative exercises. In summary, it is a magnificent compilation of ideas and techniques: it is a mine of (well-organised) information suitable for the graduate student and experienced researcher alike.' Roy Dyckhoff, Bulletin of the London Mathematical Society
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 programmes). It deliberately transcends disciplinary boundaries and challenges many established attitudes to the foundations of mathematics.
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HOW DO WE BEGIN to lay the foundations of a palace which is already more than 3600 years old? Read the first page
Front Cover | Copyright | Table of Contents | Excerpt | Index
HOW DO WE BEGIN to lay the foundations of a palace which is already more than 3600 years old? Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index
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3.2 out of 5 stars
4 reviews
11 of 12 people found the following review helpful
2.0 out of 5 stars
Useful summary, poor exposition
26 Mar 2006
By J. Elliott
- Published on Amazon.com
Format: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 15 people found the following review helpful
1.0 out of 5 stars
A confusing hodgepodge of tangential ideas
10 Aug 2006
By jasonc65
- Published on Amazon.com
Format: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.
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.
5.0 out of 5 stars
An Enjoyable and Insightful Exposition
18 Nov 2012
By Hans-Martin Will
- Published on Amazon.com
Format:Hardcover
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.
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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3.2 out of 5 stars
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