I have surveyed all Mathematica books up to year 1997. Some of them have come and gone, while a lot more Mathematica books have turned up since. Here are some of the best Mathematica books i know today (as of 2002).
The Mathematica Book By Stephen Wolfram. amazon
I've read this book systematically cover to cover at least 3 times during 1993 – 1998. If you must have only one Mathematica book, this is the one. This book is online at http://library.wolfram.com/.
Mathematica GuideBook By Michael Trott, 2004. (3 volumes) amazon
I haven't had a chance to check out these books. However, i know Michael Trott personally. Believe me, these books are the state of the art of Mathematica. Michael Trott is the world's top Mathematica programer, especially when it comes to programing graphics in Mathematica. He's probably the world's #1 expert in graphics programing in Mathematica.
The book's website is at: mathematicaguidebooks.org
Programming in Mathematica By Roman E Maeder. amazon
If you ever intent to write a Mathematica package, you need this book. Every Mathematica programer needs to read this book.
Maeder is one of the core designer for the Mathematica engine. He's got several other Mathematica books, including The Mathematica Programmer II amazon and Computer Science with Mathematica amazon. This computer science book is not very good though.
Power Programming with Mathematica (≈1997) By David Wagner. amazon
I have not read this book, but it is acclaimed and i believe it from what i know of David Wagner. This book is for professional Mathematica programers who wishes to understand Mathematica constructs and behavior and their respective performance.
The Beginners Guide to Mathematica By Jerry Glynn and Theoder Gray (3rd ed, ≈2000). amazon
A beginner's guide, exactly that. If you are a student and using Mathematica for the first time and are scared by the huge system, this is a excellent icebreaker. It gives you a bird's eye view of the entire system, and in a fun way complete with the author's own rants on mathematics software and education. Let this book ease your fear.
One of the author Theodore Gray is the author of the Mathematica interface. The part of the book on math software is online on his website at Source.
A New Kind Of Science By Steven Wolfram, 2002. amazon
In the year of our lord two thousand and two, a New Kind of Scientist Stephen Wolfram brings us a epoch-making book on Cellular Automata captioned A New Kind of Science, which engendered a brouhaha in academia, in which a renowned prodigy, accomplished cosmologist, and wildly successful millionaire and recognized megalomaniac proclaims that he has discovered a theory of everything, and so expressly tells the reader that every scientists in history are essentially morons of the first order in comparison to him.
Some will simply call the book trash, while others take a wait-and-see stance. In yours truly's humble opinion, this is a substantial book on the topic of cellular automata (if nothing else), and worth reading for math enthuses considering its relatively cheap price.
To hard scientists (most of which knew little about society and psychology): One may need to look at life and society as a process kind of view, a ever-changing dynamic system, a artificial-life model. The human-simpletons are just little insignificant entities in a environment of billions of them, each effecting local happenings in a diverse and extreme complex way with some simple but fuzzy needs, along which some emergent phenomenon arises, among them survival code, morality, and personalities. Stephen Wolfram and his bearings thus can be considered as a abnormal survival strategy, and some negative reactions thus a animal-pack's counter-strategy. Similarly, pro-Wolfram ass-kissers are thinking for themselves too. May you be a moralist, or logician, ultimately you cannot escape humanity (you need to eat, to say the least), therefore unknown to many thinkers that there is a ultimate factor that comes before truth. To be oxymoronic: all truths are human truths.
As to Stephen Wolfram's deeds… for one thing we must understand: he did create Mathematica and by that he has changed the world. Then now, he did create this "A New Kind of Science" and for good or bad he again stirred the world. If in the near future we indeed see a paradigm shift towards research of things thru computation, then we may congratulate him not because he is the originator of a idea, but that his calculated fantasies went jackpot. This is what entrepreneurism is about.
See also: Criticisms on Criticisms on a New Kind of Science.
Modern Differential Geometry of Curves and Surfaces, 2nd ed. By Alfred Gray. amazon
Not very good. This is a ugly book. Bulky and ugly prints. Nor is it a quality differential geometry book. It doesn't teach the least bit of Mathematica either. Nevertheless it is one of a kind suitable for Mathematica enthuses. If you want to see all the exotic manifold realized as computer graphics, buy this book.
In general, i'm of the opinion than math text books should not be intermingled with math software. If you are teaching math, teach math. If you are teaching math programing or software, than do that. Let the student integrate and interpret disparate things themselves; that is learning.
There are book such as Calculus with Mathematica amazon , Complex Analysis with Mathematica amazon, Abstract Algebra With Mathematica amazon , stats with Mathematica amazon , Differential Equations with Mathematica amazon, … etc., these are best used as a supplement, or for those who want quick results than true understanding.
The Mathematica Graphics Guidebook By Cameron Smith, Nancy Blachman. amazon
If you are into heavy graphics programing in Mathematica, you need this book. There is no other. This book is a bit old, however it is still relavent in 2002 for Mathematica version 4. It is probably too old in year 2004.
There is another book on Mathematica Graphics: Mathematica Graphics: Techniques ＆ Applications By Tom Wickham-Jones. amazon This is more of a graphics application book. This book is probably also outdated. If you are into Mathematica graphics programing, get Cameron Smith's book first.