Conceptual Mathematics: A First Introduction to Categories Paperback – 26 Oct 2011

Over the last two years I have revisited different sections of this book and gain new profound insights with every read. With some dedication and time, this book will surely enrich your life! What this book offers is the truth. The concepts presented in this book are the underlying unifying ideas which make up mathematics itself in an even more general and profound sense than Set Theory (in fact, one of the authors has rigorously shown that set theory is a very special case of what is presented in this book). We can encounter categories not only at the microscopic level (where we define the fundamental ideas that allow us to construct mathematical concepts from the ground up), but at the macroscopic level as well (where complex constructions in distant fields become analogous to the microscopic building blocks). With these ideas we can show that multiplication and addition are actually more appropriately opposites of one another than addition and subtraction or multiplication and division. This book is the key to beginning a journey to discovering the true nature of mathematics. To continue (or supplement) your journey, also pick up a copy of Sets for Mathematics By F. William Lawvere and Robert Rosebrugh. With time and practice (attempt the exercises from both books!!!) you will be greatly rewarded. As a student of Mathematics, this has paid off in ways I never thought possible and continues to provide insight to nearly everything I learn in school and on my own.

A startling demonstration presented in this book is that Cantor's Diagonal Argument in generalized form not only proves that there are infinite different levels of infinity, but also Godel's Incompleteness Theorem! Also contained is a convincingly appropriate abstraction of the characteristic function of any subobject with respect to any object it is contained in (in any sufficiently rich category). In other words, mappings in the context of a chosen category with domain X and a particular codomain Omega can correspond exactly with all objects contained within X. The latest Edition elaborates on this notion of parthood as well as introduces adjoint functors.

Not long ago, I spoke with a professor at strong HBCU department. Her Ph.D. was nearly twenty years ago, but I shocked her with the following statement, "Most of our beginning graduate students [even those in Applied Mathematics] are entering with the basic knowledge and language of Category Theory. These days one might find Chemists, Computer Scientists, Engineers, Linguists and Physicists expressing concepts and asking questions in the language of Category Theory because it slices across the artificial boundaries dividing algebra, arithmetic, calculus, geometry, logic, topology. If you have students you wish to introduce to the subject, I suggest a delightfully elementary book called Conceptual Mathematics by F. William Lawvere and Stephen H. Schanuel" [Cambridge University Press 1997].
From the introduction: "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 better methods for understanding and usual mathematical concepts. While the insight and methods are simple ... they will require some effort to master, but you will be rewarded with a clarity of understanding that will be helpful in unraveling the mathematical aspect of any subject matter."
Who are the authors? Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding. Though Schanuel has published research in diverse areas of Algebra, Topology, and Number Theory, he is known as a great teacher. The book is an edited transcript of a course taught by Lawvere and Schanuel to American undergraduate math students. The book was actually chosen as one of the items in the Library of Science Book Club. The concepts of Category Theory in Conceptual Mathematics are presented in the same way Lawvere and Schanuel implemented it, in a real classroom setting, addressing common questions of students (yes these are real people) at crucial points in the book.
The book comes with thirty-three Sessions instead of Chapters. Some Sessions can be understood in a single class or hour. Others may take longer. There are also numerous Examples, Problems, and five Tests of the student's understanding.
The title of Session 1 is "Galileo and the flight of a bird" and motivates the notion map. The sixth part of Session 5 is called "Stacking in a Chinese restaurant" and helps motivate sections and retractions. Session 10 motivates the Brouwer Fixed Point Theorem. Less you think this is all Abstract Mathematical nonsense, Session 15 is called "Objectification of properties in dynamical systems." The title of Session 20 is "Points of an object."
I have recommended Lawvere and Schanuel to motivated high school students. I certainly suggest this clearly written "Conceptual Mathematics" for undergraduates. I even suggest it for the mathematician who needs a refresher on modern concepts.

This a re-print of a review I wrote for the quarterly of the National Association of Mathematics.

It has flaws, but is still one of the greatest maths book I've read. Aimed at high-school level and up, but towards the end it gets a bit complicated, so I doubt if a high school kid can fully understand it without consulting other books. But, most of the book is really easy to read, and the authors' effort to write such a book is admirable.

Lawvere is one of the developers of topos theory, where he found an axiomatization of the category of sets.

The last 2 sections are an introduction to topoi and logic. One key fact seems to be missing which caused me some perplexing: In the category of subobjects, 2 subobjects A and B has A > B if A includes B. Thus, the relation ">" creates a partial order amongst the subobjects. If A > B and B < A, then A = B, thus inducing an equivalence class, denoted by [A]. This is the reason why the subobject classifier has internal structure (different "shades" of truth values).

Also, the relation of topology to logic is analogous to the relation of classical propositional logic to the Boolean algebra of sets, with the sets replaced by open sets in topological space.

I've only read the 1st edition. The 2nd edition's first part is the same as the 1st edition, with additional advanced topics at the very end.

Such an excellent work as one is given to saying asto all productions of Lawvere's. This book, on the face of it, seems easy, even elementary. But there is, as Lawvere has said, an awful lot here. A book is elegant if it achieves to say a great deal with ease and a sense of depth of coverage.

The path to Cateogories and Toposes is via two book: Cat for working mathematicians and Sheaves in Geometry and Logic both by Mac Lane. But these are anything but easy or elementary.

There is a problem with mathematical texts of a pedagogic kind, one that this book avoids: their writers often confuse teaching with forma exposition. They don't "talk" to one but go off at their own formal tangents.

I spent probably about 40 hours on this book. Its hard to judge since I read it before going to sleep, off and on, over the course of a year. The layout of the book is interesting. They divide the book up into articles and sections; with each article a shotgun introduction to the topics discussed in the following sections. I was really interested in learning the topic, and am still learning it, but I really didn't find this book too useful. It is written in a conversational style, like that of a professor teaching a small class of students. My main problem was the pace of the exposition. It often seemed very slow, belabored, and too conversational. On the other hand--near the end, when the topic turns to the exponential objects--I felt out of my league and unprepared. Since then, I've gotten through the first 100 pages of Goldblatt's "Topoi," from which I feel I gained a much better insight into category theory than I did from "Conceptual Mathematics", and in much less time. "Topoi" cheats, in a way. It develops category theory through first describing the concepts in terms of sets. This is useful for someone who is already familiar with set theory. "Conceptual Mathematics" does not do this. On the one hand this is admirable, since category theory is a fundamental theory at least as fundamental as set theory. On the other hand it ties one hand behind the authors back by witholding a descriptive tool from them that most people interested in the topic will already be familiar with. In my opinion leads to a sloggish read. "Conceptual Mathematics" just wasn't direct enough for me. That said, the book does deserve three stars; it is not a *bad* book.