This book is by far the best book on mathematics I have ever read. It teaches the concepts in an intuitive, exciting way, and yet it is able to remain fun and engaging throughout. Technical material is tackled, in depth, without there seeming to be any work done. There are no exercises to be done, you simply follow Stewart along for a tour through modern mathematics. Ian Stewart's writing is flawless and almost turns this book into a thriller. I read this book in one night- I could not put it down! I stayed up until 4 in the morning reading and rereading passages; it is truly a masterpiece. The chapters are as follows:
Chapter 1- Mathematics in General: Here Stewart describes certain aspects of mathematics, and discusses their purpose and implications. He talks about abstractness and generality, intuition vs. formalism, and pure vs. applied mathematics. He tells the reader the importance of understanding WHY a theorem is true, not simply that it is. He ends with a collection of anecdotes.
Chapter 2- Motion without Motion: This is an example of thinking a bit outside the box. The chapter is devoted to overturning Euclid's proof that the base angles are congruent, and making a new one based on rigid motions. It doesn't sound too engaging, but, somehow, Stewart manages to make it quite exciting!
Chapter 3- Short Cuts in the Higher Arithmetic: A basic introduction to number theory- prime numbers, moduli, congruences, etc. The informal tone makes this the easiest and most understandable read on number theory I've yet encountered.
Chapter 4- The Language of Sets: Throughout the rest of the book, Stewart uses the language of set theory, so he introduces that here in an easy to understand way (using some imagery like bags of items, etc).
Chapter 5- What is a function?: Here Stewart addresses some of the historical problems of defining a function, and then uses the set theory from the previous chapter to define a general function, and the different types of functions.
Chapter 6- The Beginnings of Abstract Algebra: An introduction to groups, fields, rings, etc. Stewart uses the rigid motions from Ch. 2 as an example of the group concept, and then goes on to make a proof about the game solitaire (the British version) using groups. Also an explanation of the proofs about constructibility (trisecting an angle, etc) are given here.
Chapter 7- Symmetry: The Group Concept: This is where we begin to see that Ian Stewart may have a bit of a bias towards abstract algebra and group theory, as that is his specialty. That is perfectly fine, but definitely something to be aware of. The chapter on Real Analysis is certainly less in-depth than this one, but there are many hundreds of books on that you can use to fill the gaps. (Also, Real Analysis is difficult to make accessible to those without a background in calculus, whereas algebrais concepts are fairly natural). In this chapter Stewart discusses groups, subgroups, and isomorphisms with great passion.
Chapter 8- Axiomatics: This is one of my favorite chapters, and it centers on Euclidean geometry and the importance of axiomatics. It discusses models, the parallel postulate, alternate geometries, consistency, and completeness.
Chapter 9- Counting: Finite and Infinite: This is the standard treatment of Cantor and his amazing discovery. I mostly skimmed this chapter, because I had just completed a book specializing in the subject.
Chapter 10- Topology: From Mobius strips, to Klein Bottles, to orientability, to the Hairy Ball Theorem. This chapter keeps to its title. I especially love the last line about the Hairy Ball Theorem (which is a theorem that seems entirely useless at face value). "It has one application in algebra: it can be used to prove that every polynomial equation has solutions in complex numbers (the so-called 'fundamental theorem of algebra')."
Chapter 11- The Power of Indirect Thinking: This is a foray into graph theory and Euler's Formula. A lovely discussion at the end about coloring, as well.
Chapter 12- Topological Invariants: Continues the discussion of topology and proves Euler's generalized formula. Also classifies surfaces, and proves some more coloring theorems.
Chapter 13- Algebraic Topology: You can see that topology is an incredibly important tool in modern mathematics. Here he discusses Holes, Paths, and Loops.
Chapter 14- Into Hyperspace: A short treatment of polytopes and higher dimensions.
Chapter 15- Linear Algebra: A bit on the geometrical, set-theoretic, and matrix views of solving simultaneous linear equations.
Chapter 16- Real Analysis: A light treatment of infinite series, limits, completeness, continuity, and proving analytical theorems.
Chapter 17- The Theory of Probability: Random walks, binomial distibution, etc. Treated informally.
Chapter 18- Computers and Their Uses: Programming and how it works on a mathematical level.
Chapter 19- Applications of Modern Mathematics: A very interesting read about optimization and catastrophe theory.
Chapter 20- Foundations: The best treatment of Godel's proof I have yet to see. It is surprisingly rigorous, but easy to follow.
Appendix- And still it moves...: This was added 5 years after the book was written, and is an absolute gem! Stewart addresses the proof of the four-color theorem, he talks about polynomials and primes, he talks about chaos and attractors, and he ends with a reflection on real mathematics. A great end to a masterpiece.
This book is for everyone and anyone- a modest background in high school algebra and an appreciation for mathematics is all you need. Buy this book! Give it to your friends!