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Type of bind: Paperback
Dewey Decimal Number: 516
EAN num: 9780521478175
ISBN number: 0521478170
Label: Cambridge University Press
Manufacturer: Cambridge University Press
Quantity: 1
Page Count: 376
Printing Date: November 28, 1997
Publishing house: Cambridge University Press
Sale Popularity Level: 310042
Studio: Cambridge University Press
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The idea of a 'category'--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the very first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study.
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Rated by buyers 
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I've been interested in category theory for many years, and I have several books on it. This is the very first book that is truly an introduction to categorical thinking. I found it very stimulating and understandable. Read this first, then try some of the other books.
Rated by buyers -
When I saw the one star review, and the general disagreement about the merits of this book I couldn't help but jump in. clearly this is not a work aimed at the professional mathematicians who will find it only tedious and repetitive. No, it has been designed for the many folks, pretty much like me I assume, who have some background in undergraduate mathematics and would love to learn more about the fascinating, and to them "new" field of category theory. It takes such folks by the hand and not only explains clearly the basic concepts, but, much more importantly explains WHY such seemly obvious and highly abstract concepts become exceptionally useful, and productive of insights, when applied to a huge number of ostensibly largely unrelated areas within mathematics and its many practical applications, and THAT, after all, is what Category Theory, much like Abstract Algebra and Algebraic Topology before it, is really all about. So sure, if you already know a great deal about the many general features thematized by Category Theory that are useful everywhere from Boolean Logic to AI, to Lord knows what all else, this is not the best book for you to start with -- though, provided one isn't an elitisit killjoy, it might still interest you to see how two very highly respected scholars have attempted to lay the field out in such a way that it can engage folks who are just beginning to move into such areas of mathematical abstraction. Anyway, I give it five stars, since my only reservation is the minor complaint that at times the examples -- the much maligned "word puzzles" -- aren't quite as stimulating or "right on the money" as they perhaps could be. It is also true that, provided one has sufficient mathematical background, it is actually easier to progress more deeply into the field more quikly using Steven Awodey's now already standard "Category Theory." It provides most of the same materials, and a good deal more, a bit more parsimoniously and directly, without requiring the reader to take nearly so many "detours," some of which seem rather obvious, although it is also a bit less thorough in explaining what the long term theoretical payoff will be. All things considered, however, mthematicis pedagogy yesterday still cries out for MORE, not fewer, books like "Conceptual Mathematics," and fortunately a new breed of fine authors is now beginning to work the boundaries between rigorous mathematics and clear popular presentation, a need that was barely recognized only thirty years ago, except by a few intrepid souls including Gamow, Conway, Rucker, Barrow etc.. All in all, at least IMHO, these two authors, both very first rate Category Theorists, are to be greatly commended, not condemned, for having written a work which works to make their field more readily comprehensible to others with lesser immediate preparation for it.
Rated by buyers -
The theoretical physicist John Baez wrote, "[Conceptual Mathematics] may seem almost childish at first, but it gradually creeps up on you. Schanuel has told me that you must do the exercises--if you don't, at some point the book will suddenly switch from being too easy to being way too hard! If you stick with it, by the end you will have all the basic concepts from topos theory under your belt, almost subconsciously."
Conceptual Mathematics has only two prerequisites: Basic high-school algebra, and willingness to work through the material carefully. In return, this book offers a solid introduction to Cartesian closed categories and topoi. Major topics include sections and retractions, initial and terminal objects, products and coproducts, exponentiation, and subobject classifiers.
These topics are illustrated using a variety of basic categories, each of which the authors introduce from scratch. These categories include sets, dynamic systems, and graphs, plus many variations of these categories. The self-contained nature of these examples is the book's greatest strength--almost every other introduction to category theory assumes prior knowledge of either topology, logic, or theoretical computer science.
But why take the time to study Cartesian closed categories and topoi? An example may help.
In computer science, the best-known Cartesian closed category is the lambda calculus, which lies at the heart of functional programming languages like Haskell and Scheme. But Cartesian closed categories appear everywhere in mathematics, logic and theoretical physics. And these connections between subjects can be exploited: For example, there's a program named Djinn, which translates Haskell type signatures into statements in intuitionist logic (using the Curry-Howard-Lambek correspondence). From there, Djinn runs a theorem prover, and then translates the output back into Haskell functions satisfying the original type signatures. In other words, by exploiting the connection between type systems and logic, it becomes possible to use tools from one field to solve problems in another.
A word of caution, however: Conceptual Mathematics omits several central topics in category theory, including functors, natural transformations, and adjoints. In many cases, it lays extensive groundwork for these topics, but never gets around to covering the topics themselves. So if you want to go beyond a basic introduction to closed Cartesian categories and topoi, you're going to need another book.
Despite these limitations, however, Conceptual Mathematics is an enjoyable--and uniquely accessible--introduction to category theory.
Rated by buyers -
Many of the reviews evaluate the book from the perspective of graduate students in mathematics want to learn categories, and it's certainly the wrong choice for that purpose. If you think of this as a serious math textbook, then it fails in that goal: significant proofs are the exception rather than the rule; very few, and trivial, exercises; very lacking in depth.
This is a great book because it provides a motivation for investigating categories. It helped me when I was in the position of hearing from a lot of places that subjects I was interested in often used category theory. I tried to read a few "real" books about category theory, and didn't get very far because they did not make the connections I was looking for. I accumulated three or four such books, all with bookmarks at about page 50 to 75. This book taught me relatively little about the theory of categories or the body of knowledge about them, but it provided a wealth of connections between categories and other topics, which made me better able to finish a couple of the real books and figure out what I needed to know there.
My advice, if you're in anything like that situation, is to read this book. Just don't take it too seriously, and don't try to milk more out of it than is really there. Then go learn more about category theory from elsewhere.
Rated by buyers -
In the preface of this book, the author comments that this book has been used successfully in high schools, colleges, graduate schools, and by professors. After reading this book, I can believe it. This book is simply a gem.
Mind you, although this book is very easy to read, some of the concepts contained within it are very abstract and can be very difficult to fully comprehend. While a high school student will surely get something out of this book, it would be hard to understand everything in it without knowing a fair amount of mathematics.
I would recommend this book to any mathematician. It is an absolute must-read. The author makes the claim that working through this book will improve your ability to categorize (no pun intended!) your mathematical knowledge so as to better know how to approach problems. From my experience, this claim is true. This book somehow teaches some of the things about problem-solving that many people believe cannot be taught.
This book looks deceptively simple, especially relative to beasts such as MacLane's "Categories for the Working Mathematician". However, I find that I keep coming back to this book, sometimes after several months. In particular, I have found that reading this book has opened the door to understanding some of the advanced mathematics books that previously seemed inaccessible to me, such as Lang's "Algebra".
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