Rather than simply throw mathematical puzzles and diversions at us, or weird and wonderful numbers, Eugenia Cheng takes us on something close to a journey through the mathematical mind, introducing us first to abstraction, then through the processes of mathematics, the way it generalises and the essential foundations of axioms. This is all as an introduction to the second half the the book on Cheng's speciality, category theory, which will I suspect be as unfamiliar to most non-mathematicians as it was to me.
So in terms of what it sets out to do and what, to some degree, it achieves it is absolutely brilliant. Cheng writes in a light, engaging fashion and really pushes the envelope on the way that you can explore mathematics. The basics are there - the inevitable doughnut/coffee cup topology comparison (though she prefers bagels, as doughnuts are not always toroidal), for instance, but this quickly then evolves into the much more challenging concept of 'taking the complement' of something by removing it from three dimensional space with an imaginary three dimensional eraser and examining what remains through topological eyes.
I can't totally ignore the issues. The lesser one is that as a gimmick, each section begins with a recipe which is then used to illustrate a mathematical point (though also to talk about food) - I found this a touch condescending and very irritating, though some readers will probably like it. The bigger problem is that the author isn't great at structuring a book. The first chapter particularly is all over the place, and she has a tendency to use concepts before they are explained. This is particularly true of category theory, which never really gets a clear, approachable definition, but rather is feinted at to begin with, and then introduced as example after example, which without a structure explaining just what it does is quite difficult to put together as a total picture of a discipline.
So, flawed it certainly is, but that doesn't get in the way of it being an unusually interesting attempt at doing something far more significant than most popular mathematics books do. I've always felt that pure maths was uncomfortably abstract and arbitrary, coming up with rules that have no obvious justification. This is the first book of read where it's possible to get a sense of, 'Hey, that kind of makes sense' - which surely is an impressive achievement. If you can look past the gimmicky aspect and the occasionally confusing structure you are in for a treat.
Review by Brian Clegg