I spent another hour on Irreligion this morning, and I've decided not to pursue it any further. There's an adequate review on the New York Times website (warning: the review contains a reference to a visual image from the book that some readers may find disturbing), and if I may summarize, I think Jim Holt basically says what I said before: John Allen Paulos doesn't take his subject seriously enough to do a good job.
Most of the chapters follow a pattern where Paulos outlines an argument for the existence of God, summarizes it in a few bullet points, and then discusses a mathematical idea that is at least superficially similar, and suggests that since the mathematical idea provides us with knowledge then there's some problem with the way the bullet points reach their conclusion. He does this with varying success; some of his arguments are stronger than others. I'll refer interested readers to the book rather than treat particular chapters ham-handedly.
In the end, though, I wasn't even satisfied with the mathematics itself; while I'm sure, for example, that it's difficult for a random medium not to have pockets of apparent order, I'm not sure that's applicable to the universe in which we live: the numbers end up being too vast. And while I find it easy to believe that branching probabilities indicate that family trees either rapidly die out or expand to fill the space allotted to them, I'm not sure that disproves the notion that someone who lived two thousand years ago couldn't have just one descendant today. And sadly, Paulos seems to play fast and loose with familiar and established notions of probability: yes, when we shuffle a deck of cards each outcome is equally unlikely, but what does that have to do with evolution?
I think the problem I had with the book was that it was short on mathematics in addition to being short on theology (or whatever arguments for the existence of God are called nowadays). There's only so much "these two lightly-described things are superficially similar, therefore I can apply one of them to the other" I can take.
Sadly when I was done this book cast some doubt on how useful the other Paulos book I'd read (A Mathematician Reads the Newspaper) is. I can take some consolation from the idea that mathematics is good for making ax-like cuts (there are no perpetual motion machines; a 3% real annual return will not make you fabulously rich) but the greater the precision required the more difficult it is to apply mathematical models to familiar problems.
exercises in compound storytelling