The Oxford Users' Guide to Mathematics is an ambitious reference book. It strives to be readable by highschoolers, graduate students, mathematics researchers wishing to learn about other fields of mathematics, and professionals wishing to learn about how mathematics applies to their chosen fields. This aim raises two questions for the reviewer: what mathematics does it contain and does the book succeed in being readable across such a broad spectrum of users?
I've not spent much time with mathematical reference books in my career; basically, I think of them as containing tables of integrals and various other formulae. You'll find those here — in Chapter 0. What else will you find at your fingertips in this book? Well, what do you want to know about analysis? Look in Chapter 1. At almost 400 pages, there's quite a lot here, from elementary facts about real and complex numbers, in the first few pages, to applications of complex functions to hydrodynamics and electrostatics in the last section. Chapter 2 covers topics in algebra, and chapter 3 geometry, including sections on algebraic geometry and the geometry of physics (Minkowski space, spin groups, sympectic geometry). And so on. (See the table of contents .) This isn't an exhaustive book (as a topologist, of course I wonder why topology is missing), but it does contain quite a bit.
To speak to the level of the text, consider the topics of Chapter 1 in more detail. This chapter focuses on analysis. The first section lays out the properties of numbers, reminding the reader of how to work with fractions, how to work with decimals, and how to work with complex numbers. After taking the care to do this, it seems surprising that I did not find a defintion of the term function when I first encountered functions here. Checking the index did not reveal where a definition of this term might be found, although the earliest sub-term mentioned led me back to the middle of a whole section in Chapter 0 about functions. This isn't a book I'd expect to read sequentially, however, except within chapters. From here the contents of Chapter 1 resemble a basic calculus sequence followed by a crash course in partial differential equations and complex analysis. The topics are tightly woven, and include, for example, exposure to metric spaces, Fourier transforms, and differential forms to name a few of the more advanced concepts. The author does provide context; many sections begin with "basic ideas" and overviews.
To sum up, this book succeeds reasonably well at achieving its stated goal. I expect that readers at the low end of mathematical background will find the text helpful for its collection of formulae, and for a quick tour of the theory of calculus etc, but will be rather overwhelmed in general. Researchers in mathematics may be underwhelmed. Those who studied some mathematics at one time, and who would like to learn a bit more of the theory or brush up in some areas will probably be the most pleased with this reference.
Michele Intermont teaches at Kalamazoo College.