Conceptual Mathematics: A First Introduction to Categories

I read it as a first year pure math undergraduate, and though it was at times at too low a level (the 'tests,' for instance, are very easy reviews of basic ideas), it never became boring. For me, it read 'like a novel' (and a page-turner, at that). My only gripe is the lack of an annotated "further reading" section, which would have rounded out the book.


Intuitive Introduction   June 6, 2000
20 out of 20 found this review helpful

Highly intuitive introduction to this abstract, but highly practical area of mathematics with one glaring fault. First the good news. I have never seen a more carefully explained introduction into an area of mathematics. Many examples and explanations of the principles behind and applications of concept analysis. However, the glaring fault is organization. Details are given without adequate tie in to how they relate to others. The text bounces from one area to the next so it is easy to lose sight of the whole picture. On balance its strengths far outweigh its weaknesses so I recommend it without reservation.


Objects and maps are everywhere   June 21, 2004
Dmitry Vostokov (Monkstown, Co. Dublin Ireland)
13 out of 14 found this review helpful

Excellent book for non-professional mathematicians, like me (I'm a software engineer), who wants to understand modern mathematics and apply its ideas in analysis of complex problems. Lots of pictures and diagrams (compared to terse wording in other mathematical books) really help to understand and master the subject. I think most of negative reviews come from professional mathematicians, but they don't need this book.


Oustanding book: an absolute must-read for any mathematician   October 21, 2006
Alexander C. Zorach (New Haven, CT)
9 out of 9 found this review helpful

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".


One Star? Such a Generous Heart!!   August 7, 2008
William D. Fusfield
1 out of 1 found this review helpful

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 today 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 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.