Conceptual mathematics: A first introduction to categories

by F. William Lawvere

In the last fifty years, the use of the notion of 'category' has led to a remarkable unification and simplification of mathematics. Written by two of the best-known participants in this development, Conceptual Mathematics is the first book to serve both as a skeleton key to mathematics for the general reader or beginning student and as an introduction to categories for computer scientists, logicians, physicists, linguists, etc.

Genres: Mathematics, Nonfiction, Science, Computer Science, Logic, Textbooks, Programming

347 pages, Unknown Binding

First published November 28, 1997

Reviews

[Walter · Rating 4 out of 5]

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.

[M · Rating 5 out of 5]

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!

[Úlfar · Rating 5 out of 5]

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.

[King · Rating 4 out of 5]

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

[Chris · Rating 4 out of 5]

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.

[tyler]

i read this and did the problems but did i really learn anything? unclear

[Eric Tao · Rating 4 out of 5]

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.

[Mark Seemann · Rating 3 out of 5]

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).

[Carter · Rating 2 out of 5]

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.

[Didier "Dirac Ghost" Gaulin · Rating 3 out of 5]

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.

[DJ]

low level intro to category theory that is uniquely accessible to undergrads

[Dave Peticolas · Rating 3 out of 5]

My first attempt to understand what the Haskell folks are really up to. I have a feeling many more attempts will be required!

[Chris · Rating 4 out of 5]

Definitely the most accessible introduction to category theory in existence.

[Lulu]

the 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