Reviews elsewhere on the web:
Observatory magazine

John Barrow

Pi in the Sky

Pi in the Sky by John Barrow is really a combination of two different books. The first is a history of counting from the earliest times. The second is a look at the ideas of Cantor and Göel and their implication for mathematics. I can see the two parts are connected - inventing infinities is no different to inventing zero, or 1,2,3 for that matter - but Barrow doesn't really bring out this connection. The book is a bit philosophical, but it's easy to read an so is suited to those readers who want to find out more about the philosophy of mathematics without things becoming too technical.

One problem with popular mathematics books is that they all tend to deal with the same few subjects - incompleteness, codes, 'Game of life' etc. and to some extent Pi in the Sky follows this path. However, it does have some interesting material on L.E.J. Brouwer, which would be useful for those who want to find out more about intuitionism.

One criticism I have of the book is that Barrow believes in too strong a form of undecidability, for instance his claim that it is never possible to know if a computer program is the shortest one possible for it's task - this just isn't true.

Product Description
John D. Barrow's Pi in the Sky is a profound -- and profoundly different -- exploration of the world of mathematics: where it comes from, what it is, and where it's going to take us if we follow it to the limit in our search for the ultimate meaning of the universe. Barrow begins by investigating whether math is a purely human invention inspired by our practical needs. Or is it something inherent in nature waiting to be discovered?

In answering these questions, Barrow provides a bridge between the usually irreconcilable worlds of mathematics and theology. Along the way, he treats us to a history of counting all over the world, from Egyptian hieroglyphics to logical friction, from number mysticism to Marxist mathematics. And he introduces us to a host of peculiar individuals who have thought some of the deepest and strangest thoughts that human minds have ever thought, from Lao-Tse to Robert Pirsig, Charles Darwin, and Umberto Eco. Barrow thus provides the historical framework and the intellectual tools necessary to an understanding of some of today's weightiest mathematical concepts.