The first category he calls the "Classical Arguments". First he deals with the argument from a first cause. Paulos points out that if everything has a cause, then that statement includes God as well. I think some believers would reply that the premise is that all physical things have causes, but God isn't physical. I think such assertions are baseless. Paulos also makes the important point that the term "cause" only makes sense if time is involved. A causes B only if A is before B. I've heard William Lane Craig say that God's first cause was simultaneous with the beginning of the universe, but I find that idea to be nonsensical.
He also covers the argument from design making an analogy to our complex free market economy that has emerged with no central planning. Finally he goes over the anthropic principle and ontological argument. Paulos was not impressed with either of them.
The second category was subjective arguments like coincidence, prophecy, personal experience and miraculous intervention. As to be expected, Paulos finds the evidence lacking. One interesting bit of information regarding prophetic testimony is that "...testimony that someone is telling the truth is self-undermining if the likelihood of truth-telling is less than 1/2. If people are confused, lying, or otherwise deluded more often than not, then their expressions of support for each other are literally less than worthless." Paulos goes on to prove this mathematically.
In one interlude between the main arguments Paulos talks about Jesus and says how surprising it is that people take the stories about Jesus in the Bible at face value. He compares it to recent events like the JFK assassination or Watergate which were covered in detail by the modern-media with recordings on film and tape, yet we are still clueless about so much of what was going on with those historic events. Paulos also discusses the silly idea in the Da Vinci Code story that a single family descends from the line of Jesus. He shows mathematically how if a person from 2000 years ago has any descendants alive today they must number in the millions.
The final category is called psycho-mathematical arguments. Here he talks about the arguments from redefinition, complexity, cognitive tendency, universality and gambling (aka. Pascal's Wager). There is a lot of good stuff here, but I'll just end with this fascinating excerpt:
[Researchers] exposed fourth- and fifth-grade students to a variety of intriguing mathematical games and measured the time the children played them. They found that the children seemed to possess a good deal of intrinsic interest in the games. The games were fun. After a few days, however, the psychologists began to reward the children for playing; those playing them more had a better chance of winning prizes offered. The prizes did increase the time the children played the games, but when the prizes were stopped, the children lost almost all interest in the games and rarely played them. The extrinsic rewards had undercut the children's intrinsic interest. Likewise, religious injunctions and rewards promised to children for being good might, if repudiated in later life, drastically reduce the time people spend playing the "being good" game. This is another reason not to base ethics on religious teachings.If you're tired of reading all the recent atheist books (as I was) I highly recommend this little book with it's refreshing perspective on these old arguments.