At first glance, John Barrow's latest book looks like a derivative attempt to cash in on the success of such recent books as Peter Beckmann's A History of Pi and his own The Book of Nothing. But don't be fooled. Whatever the author's motive, the end product is a delight. I have read all Barrow's books, and this is the one I have enjoyed the most. Others are more erudite, or more comprehensive, or superior in their own special ways. But this one is definitely the most readable, and the most fun.
Barrow ought to know a bit about infinity, since he is a Professor of Mathematical Sciences at Cambridge. The conceit around which the book is constructed is to look at the nature of infinity in many different guises. There's the mathematicians' infinity, where an infinite number of decimal numbers exist between any two decimal numbers you choose; the physicists' infinity, where the idea of infinite space is counterbalanced by the idea of infinite subdivision of matter; the philosophers' infinity, where everything recurs infinitely many times and in infinitely many places in an infinite universe; and more besides.
One of my favourites (an old chestnut to mathematicians, but probably a new delight to most of Barrow's readers) is the Hotel Infinity. Even though all of its infinite number of rooms are occupied, space can always be found for an infinite number of new guests, and then another infinite number of new guests, and so on for ever. There's the idea that some infinities are bigger than other infinities, an explanation of why the sky is dark at night, and some serious stuff suggesting there must be something fundamentally wrong with theories of physics (including Einstein's general theory of relativity) that allow for the existence of so- called singularities inside black holes and at the birth of the universe.
When he wants to, Barrow has a light touch, and he wants to throughout this book. He offers a proof that the existence of non- zero interest rates is evidence that time travel does not happen, discusses the possibility of carrying out an infinite number …