Conceptual Mathematics: A First Introduction to Categories (Paperback)

~ (Author), Stephen Hoel Schanuel (Author) "Our goal in this book is to explore the consequences of a new and fundamental insight about the nature of mathematics which has led to..." (more)
Key Phrases: irreflexive graphs, naked dot, retraction theorem, United States (more...)

Editorial Reviews

Review

"Conceptual Mathematics provides an excellent introductory account to categories for those who are starting from scratch. It treats material which will appear simple and familiar to many philosophers, but in an unfamiliar way." Studies in History and Philosophy of Modern Physics

Product Description

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 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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Product Details

• Paperback: 376 pages
• Publisher: Cambridge University Press (November 28, 1997)
• Language: English
• ISBN-10: 0521478170
• ISBN-13: 978-0521478175
• Product Dimensions: 9.5 x 6.7 x 0.9 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.0 out of 5 stars  See all reviews (14 customer reviews)
• Amazon.com Sales Rank: #774,162 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

49 of 50 people found the following review helpful:

5.0 out of 5 stars Interesting and Accessible at Many Levels, August 2, 1998

By A Customer

Lawvere and Schanuel have created a book at once accessible and stimulating at a great many levels. It discusses the concepts of Category Theory in a simulated "classroom" setting, addressing common questions of students at crucial points in the book. It also wanders in a care-free manner through an amazing number of topics. The book is interesting to non-mathematicians at a philosophical level, and to (beginning) mathematicians as an introduction to an exciting new area of mathematics. The authors have a great attitude, and offer great starting-points for investigation.

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.

33 of 36 people found the following review helpful:

3.0 out of 5 stars Very uneven, but still useful, November 10, 2002

By	T. Gwinn (USA) - See all my reviews (REAL NAME)

As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics.

Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here.

There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together.

Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful.

Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth.

Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples. I think this will leave many readers less than satisfied, but realistically, the book would have been twice as long had they really delved deeper into examples (or they would have had to be very terse in the actual descriptions of category theory, which is the choice most authors writing for a more mathematically-inclined audience seem to make - e.g., _Mathematical Physics_ by Geroch (good book!) or _Basic Category Theory for Computer Scientists_ by Pierce).

If you are mathematically astute, you probably will find this book tedious. But if you are not a grad+ math major, then this book may well be worth the effort as a way to begin to learn a very profound and powerful set of tools and concepts.

22 of 23 people found the following review helpful:

5.0 out of 5 stars Intuitive Introduction, June 6, 2000

By A Customer

This review is from: Conceptual Mathematics: A First Introduction to Categories (Hardcover)

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.