Customer Reviews

Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics)

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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.

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.

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!