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Conceptual Mathematics: A First Introduction to Categories
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of ‘category’ for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories.
383 pages, Paperback
First published November 28, 1997
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Displaying 1 - 14 of 14 reviews
February 6, 2017
Many people think of mathematics as the operations like addition, subtraction, multiplication or division, or the complicated models used in calculus, linear modeling or differential equations. But mathematics embodies conceptual tools that are as important to understanding math as any other branch of the science. In this work, the authors lay out the concepts of conceptual mathematics in a way that is very understandable to students and to self-learners. Conceptual mathematics is sort of the bridge between philosophical logic and math, so the student is exposed to concepts much more than operational mathematics. Such concepts as the associative theorems, distributions and other items of set theory are discussed.
This book is a very good introduction to many of the concepts of conceptual mathematics that many students pick up as they study other areas of math, such as algebra or geometry. Because these concepts are hidden behind other, more specific operations, the student never really learns the concepts as such. I would imagine that many people who think that they are not good at math are simply lacking the conceptual ideas that are taught in this book. It would be interesting to teach these concepts implicitly to a group of adults who hate math and see if they make mathematics more understandable to them. If so, then this topic really should be taught to more people who feel that they don't have the aptitude to master math.
Overall I would recommend this book to anyone who wants a better understanding of the conceptual underpinnings of math.
This book is a very good introduction to many of the concepts of conceptual mathematics that many students pick up as they study other areas of math, such as algebra or geometry. Because these concepts are hidden behind other, more specific operations, the student never really learns the concepts as such. I would imagine that many people who think that they are not good at math are simply lacking the conceptual ideas that are taught in this book. It would be interesting to teach these concepts implicitly to a group of adults who hate math and see if they make mathematics more understandable to them. If so, then this topic really should be taught to more people who feel that they don't have the aptitude to master math.
Overall I would recommend this book to anyone who wants a better understanding of the conceptual underpinnings of math.
January 21, 2022
A great introduction to category theory with well thought out explanations. This book is great for a stepping stone into the ideas of conceptual mathematics and really blurs the linear nature of mathematics beginners are used to.
Understanding the power of categories is a must for any mathematician and highly recommended
Understanding the power of categories is a must for any mathematician and highly recommended
June 20, 2016
A real gem. More than an introduction to categories, if you stick with it this is an introduction to topos theory, and more generally an invitation to Lawvere-space. In other words, the treatment is largely synthetic (as opposed to analytic). Brouwer's fixed point theorem is a lovely payoff 1/3rd of the way through, and the Lawvere fixed point theorem that comes later, even better – if the treatment of dynamical systems etc. wasn't a thrill enough for the reader of what is in some respects a thoroughly elementary book. The philosophy and power of categorical thinking is also made abundantly clear. Towards the end (eg. section on connected component functor), the presentation gets a little sketchy and would benefit from supplementation from other sources, but overall a model of pedagogy and highly recommended!
January 17, 2015
Great book on category theory with well thought out explanations. It came up in Amazon recommendations when I was browsing for Haskell books and I thought I would give it a try. It was an enlightening read. I finally understand the pure mathematical power of category theory after reading this book.
December 31, 2018
as category theory books go, this is the best introduction.
Category theory will not make you into a superhero, like many people seem to think. It's just a way of organizing things.
Category theory will not make you into a superhero, like many people seem to think. It's just a way of organizing things.
August 17, 2020
CAVEAT: I completed through the sections contained in the first edition. There are some additional sections at the end of the 2nd edition.
This is a great book, but I do think there are some problems. I'm not going to talk about the good points, of which there are many, and I think this book is appropriate for anyone interested in category theory without enough background for the heavy-duty grad school versions in homology/AT/AG/whatever. Even in that case you might find something interesting.
1. I'm not in love with the structure of the book into Articles & Sessions. Particularly the "dialogue" parts. Good idea, poor execution. The questions the students asked/answers weren't really similar to the questions I myself was wondering. Example: In one section, the student and the teacher talk half a page about the appropriate wording of a definition ("No, you should say this before that").
2. I wish there were more "real" examples. The text was very heavy on dialogue, but a lot of the examples in the text didn't seem helpful toward solving the problems. In a book pitched at this level, it seems like there should be more examples and less other fluff.
3. A lot of the exercises seemed to be hinting towards more general results or examples, but it is doubtful someone reading this text has the background to realize what those were, so it would be nice to have asides (or at the very least references) to such "leading questions".
4. There's a lot of repetition, which I'm not sure if it's helpful or hurtful.
Despite my gripes, I thoroughly enjoyed the book and learned a lot.
This is a great book, but I do think there are some problems. I'm not going to talk about the good points, of which there are many, and I think this book is appropriate for anyone interested in category theory without enough background for the heavy-duty grad school versions in homology/AT/AG/whatever. Even in that case you might find something interesting.
1. I'm not in love with the structure of the book into Articles & Sessions. Particularly the "dialogue" parts. Good idea, poor execution. The questions the students asked/answers weren't really similar to the questions I myself was wondering. Example: In one section, the student and the teacher talk half a page about the appropriate wording of a definition ("No, you should say this before that").
2. I wish there were more "real" examples. The text was very heavy on dialogue, but a lot of the examples in the text didn't seem helpful toward solving the problems. In a book pitched at this level, it seems like there should be more examples and less other fluff.
3. A lot of the exercises seemed to be hinting towards more general results or examples, but it is doubtful someone reading this text has the background to realize what those were, so it would be nice to have asides (or at the very least references) to such "leading questions".
4. There's a lot of repetition, which I'm not sure if it's helpful or hurtful.
Despite my gripes, I thoroughly enjoyed the book and learned a lot.
January 5, 2024
A pretty nifty book introducing category theory to pretty much anyone interested. It starts off concrete and simple but then eventually gets into universal properties, map objects, and toposes at the very end. The only real prerequisite is interest in the subject. The structure of the book is not entirely clear to me—it has some redundant sections, which ended up being helpful for reviewing the material. Lots of nice pictures and diagrams.
January 31, 2018
The first 100 pages or so I really enjoyed, but after that, the book gradually became increasingly difficult to follow.
It seems clear that it's written by two authors; it consists alternatingly of 'articles' and 'sessions', and the sessions are much easier to follow than the articles. Even so, as the text advances, it becomes clear why Category Theory is also known as Abstract Nonsense (although I do realise that there's supposedly no negative charge in that term).
It seems clear that it's written by two authors; it consists alternatingly of 'articles' and 'sessions', and the sessions are much easier to follow than the articles. Even so, as the text advances, it becomes clear why Category Theory is also known as Abstract Nonsense (although I do realise that there's supposedly no negative charge in that term).
December 9, 2019
This illustrates the key concepts of category theory in a lot of detail. Unfortunatey it is my second book on category theory so I would actually have preferred a more succinct treatment of the theorems and concepts. For me some of the illustrative diagrams became somewhat redundant as well as a lot of the text which explained a lot of elementary mathematical concepts I have already gleaned from other sources. I have some regrets in buying this.
February 10, 2023
A quick, non formal introduction to the conceptual ideas behind category theory and the intellectual revolution that emerged from it in the domain of mathematics but also computer science and now most of the other technical fields in STEM. Intuitive and smart. A great work to accompany Category Theory for the working mathematician.
Want to read
April 14, 2009low level intro to category theory that is uniquely accessible to undergrads
October 8, 2014
My first attempt to understand what the Haskell folks are really up to. I have a feeling many more attempts will be required!
September 21, 2014
Definitely the most accessible introduction to category theory in existence.
Want to read
May 4, 2023the more I learn, the more I laugh at my previous pathetically naive ideas of what math is "all about". But if you want a glimpse, try these books
Displaying 1 - 14 of 14 reviews