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Topoi: The Categorial Analysis of Logic (Dover Books on Mathematics), by Robert Goldblatt, Mathematics
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A classic introduction to mathematical logic from the perspective of category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Its approach moves always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally.
Beginning with a survey of set theory and its role in mathematics, the text proceeds to definitions and examples of categories and explains the use of arrows in place of set-membership. The introduction to topos structure covers topos logic, algebra of subobjects, and intuitionism and its logic, advancing to the concept of functors, set concepts and validity, and elementary truth. Explorations of categorial set theory, local truth, and adjointness and quantifiers conclude with a study of logical geometry.
- Sales Rank: #290653 in Books
- Published on: 2006-04-28
- Released on: 2006-04-28
- Original language: English
- Number of items: 1
- Dimensions: 8.48" h x 1.14" w x 5.46" l, 1.31 pounds
- Binding: Paperback
- 592 pages
About the Author
Robert Goldblatt is a Professor of Pure Mathematics at the Victoria University of Wellington, New Zealand, and a Fellow of the Royal Society of New Zealand. He has served as the Co-ordinating Editor of the Journal of Symbol Logic and has been a Managing Editor of Studia Logica for the past two decades.
Most helpful customer reviews
60 of 63 people found the following review helpful.
Clear, concise, and well motivated.
By Trenton F. Schirmer
I have only read the first seventy pages of this book, but so far it is excellent. The character of the book can be likened to Kleene's "Introduction to Metamathematics" or "Mathematical Logic." The subject is well motivated by brief philosophical discussions, but the philosophy is not allowed to interrupt the flow of mathematical development. There are few exercises, but enough for the casual reader to check from time to time that he or she is understanding the material. The exposition itself is perfectly clear and concise, by my lights. Nothing important is skipped--one does not need to fill in the blanks in proofs--yet the book moves at a good pace. The focus of the book is on the development of Category theory in relation to its applications in logic. For this reason it is probably of more interest to a logician or a philosopher than to a computer scientist. Indeed, for the logician who is interested in Category theory this book cannot be missed. To put it bluntly, Goldblatt strikes me as a superior mind, even his brief remarks on set theory were enlightening. The book has been (to this point, at least) a very enjoyable read.
32 of 36 people found the following review helpful.
Enjoyable, though not as accessible as the author might hope.
By A.E.V.
I came to this book with an interest in philosophy of mathematics, formal semantics, and non-classical logics, attracted to the author's stated goals to make category theory and its implications available to both the mathematically and philosophically inclined by requiring very little background in set theory or mathematics. I have a fairly good grasp of first order logic, and know enough set theory to talk about things like Kripke semantics in a general fashion, and I find this book challenging. The author defines things like a limit on a diagram (fairly early in the book) and doesn't make it clear to me what the significance of the concept is supposed to be.
I do still find it full of suggestive and interesting ideas, and the author is clearly knowledgeable and enthusiastic. A general picture does emerge in the less formal discussions at the beginnings of the chapters, so flipping around a little can help you get an idea for the context. But ultimately I think this is still more accessible to graduate level math students, and I'm giving it a four star review because so far as I can tell, it succeeds in its technical aspects, and mainly fails only in reaching its entire stated audience.
130 of 131 people found the following review helpful.
Great introduction to the theory of toposes.
By Paul Corazza
When Goldblatt's book first hit the stands, it was blasted by reviewers who had a geometric predilection. They claimed that Goldblatt had trivialized this essentially rich geometric subject by giving a set-theoretic treatment.
I became fascinated with topos theory in the summer of 1989. My only experience with category theory at that time was some dabbling that every graduate student gets in studying modules and in functional analysis. I didn't have much background in algebraic topology where the subject is usually developed considerably further.
Frankly, I found Goldblatt's "gentle" approach to categorical concepts and the concept of a topos very rewarding. I worked through most of the book in a summer, and was fully prepared to take on the more advanced texts at that point. Later, I had the opportunity to teach topos theory at the graduate level to people with backgrounds similar to mine (i.e., without a strong background in category theory). We worked through most of Goldblatt's book in the first semester and I guarantee that all the students were very grateful, as I had been, for Goldblatt's approach to the subject.
The book is well written, accessible to graduate students, filled with fun and often challenging exercises, and packed with references. In my opinion, it is the right place to start if you want to become proficient in topos theory (and you don't already have significant proficiency in category theory).
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