Permission to Believe by Rabbi Lawrence Kelemen is a disappointing book on two fronts. Its brevity does not allow the rabbi to pursue the topic at hand with nearly enough depth and scope. Second, and especially in the argument by design, he uses science of dubious methodical soundness to prove that evolution by natural selection is statistically impossible.
These are the major flaws. A minor, and strange admissions is not a single mention of Jewish philosophers and theologians who have used proofs of God’s existence in their work. Kelemen does not cite Saadia Gaon, Maimonides, ibn Paquda. I’m not sure why you would write a book about rational proofs of God’s existence and not cite these worthy thinkers. He also ignores the Ontological Proof of God’s existence, arguably the most power of rational proofs, perhaps because it has always been in the domain of Christian theologians. This is unfortunate, since it denies the reader access to a strong a priori case for God’s existence.
The final section is the most unsatisfying of all. Kelemen uses Jewish history to prove the existence of God. God exists because Jews continue to survive despite the odds. Other ancient peoples are gone from the stage of history, but Jewish people march on. This is certainly an argument for God’s existence, but not strictly a rational one. It falls more in the faith category, and should not be in this book.
Unfortunately, a very good book on Jewish work on the rational proof of God's existence has not been written (that I can see). Certainly, this is not it.