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.