Like Adin, one of the characters in this book, many people initially become infatuated with mathematics because it seems to be a doorway to absolute truth and absolute certainty through pure reasoning. The book does a good job of presenting this viewpoint so that those who have never considered it this way can begin to have this sort of appreciation for mathematical logic. However, as with many infatuations, we soon learn that mathematics is not quite as perfect as we would like. It could be argued that mathematics still may be as close to truth and certainty as humans can get, but it certainly falls short of "absolute". Instead, it offers only a relative truth (relative to a choice of axioms) and near certainty (subject to the possibility that the axiomatic system itself is inconsistent). Again, the book does a great job of presenting this while still leaving us with an appreciation for mathematics despite its shortcomings, all the while forcing us to consider the implications for theology and the human condition.

This book is very mathematical. We essentially sit in on Ravi's "infinity" class, learning about Zeno's paradox, convergence of infinite sums, Cantor's theory of transfinite cardinals, Zermelo-Fraenkel axioms of set theory and the independence of the Continuum Hypothesis. Similarly, the jailhouse conversations of his grandfather and the judge mostly focus on the axiomatic method of mathematics, especially Euclidean geometry (and later non-Euclidean geometries as well). But, the continual focus on the comparison between these mathematical methods and religion, and on questions of certainty in our daily lives, keep it from being just another mathematics textbook. In other words, it helps the reader to see the human/emotional side of mathematics that many mathematicians feel but do not properly convey to non-mathematicians.

Interestingly, from reading the plot summary, I assumed that Gödel's Theorem was going to be the main topic of discussion in the book. In fact, it is mentioned and holds a place of special honor, but it is certainly not the focus of the mathematical discussions. I must say that I was relieved by this since Gödel is practically cliche in mathematical fiction. Interestingly, the book barely touches on the idea that mathematics might actually contain contradictions, which is one of the unavoidable consequences of Gödel's work. (See, for example, Division by Zero). This possibility is why I said above that we are reduced to "near" certainty, even when discussing truths in pure mathematics, since we must acknowledge the possibility that mathematics is fatally flawed. Instead, this book emphasizes the existence of undecidable statements.

The authors, childhood friends who each hold a Master's Degree in mathematics, do a very nice job of tying some deep mathematics together in a story that maintains the reader's interest. Actually, although it is quite clever how the mathematical subplots parallel the main story, I almost feel that the parallel is too perfect. The carefully constructed didactic elements made it more difficult for me to suspend disbelief and believe that the story was really happening.

I am sure that some readers will have trouble getting through some of the mathematical passages, which can be a bit dense if you don't already know the concepts, but I think that the book will still work if the reader simply tries to do their best with these passages and skims ahead when necessary.

I do believe there are a few mathematical errors in the book. These small mistakes do not detract from the literary quality of the book, but given the nature of this site, I thought I should point them out. (Of course, it could be that it is I who am mistaken, since the authors clearly know mathematics quite well. If that is the case, please correct me.) For instance, in the Euclidean proof that a line of a given length can be "cloned" in another location, the label "H" seems to mysteriously move from one location to another. Also, when discussing maps from a set to its power set, the examples given seem to be maps to the power set of the power set instead.

As for the philosophy, I think no mathematician could argue with their conclusions about mathematical truth being relative, that is, depending on the choice of axioms. They correctly point out that the interpretation of this varies from person to person, with some mathematicians believing still that there is a "mathematical truth" which goes beyond what we are able to reproduce with axioms and others believing that mathematics is simply a product of the human mind with no independent existence. (And, of course, there are those whose views lie somewhere in between.) I agree with the authors' analogy between this dichotomy and the question of the existence of God.

On the other hand, I think both atheists and theists will be disappointed with the apparent conclusion of the book that, in the absence of certainty, any "axiom" is as good as any other. (In the context of religion, it seems to suggest that whatever assumptions about God you begin with you might as well just keep believing them.) Let me address this using mathematics rather than religion as the subject of discussion.

The way I think of mathematics (and this is completely in line with the presentation in the book) is that it is a way for us to create imaginary, "toy universes". When we come up with axioms, it is like we are choosing the laws that will govern that universe. Then, through our investigations (i.e. proving theorems) we can learn what happens in that universe. If there is a contradiction or if there are no interesting theorems, then that axiomatic system is not worth studying. We look for rules that make for interesting results and that is what is published in "pure" math journals. But, when we are doing any kind of applied math, then it is no longer true that one axiomatic system is as good as any other. Now, the axioms have to be selected so as to be in agreement with what is known about the real world when doing applied mathematics. This may not completely determine the axioms (we really don't know everything), but it probably can exclude a bunch. I would argue that the same is true for religious beliefs and other "life axioms" as well: although we can never be certain of our choice of axioms, observations of the world around us can give us reason to abandon one set of axioms in favor of another set that are in greater agreement with reality. In other words, although the book seems to suggest that you might as well just accept your "axioms" and go with it, I would argue that you should always be willing to challenge your assumptions and throw them out when they lead either to internal contradictions or when some "theorem" they produce disagrees with reality.

An unusual literary device used throughout the book is to present ideas in the form of journal entries from famous historical mathematicians including Euclid, Riemann, Gauss and Cantor. Although effective, this may confuse some readers who mistake these for actual quotations from writings of those mathematicians. The authors make it clear in the foreword that this is not the case, but not everyone reads the foreword! As they explain, the motivations attributed to the mathematicians in these passages are fictional creations and the language/notation used is often very different than what the real mathematicians would have used at the time.

The treatment of a Hindu mathematician by the Christian inhabitants of (fictional) Morisette, NJ, and the conditions in British controlled India are among the other subjects addressed in the book. Also of interest is the character of Claire, a romantic interest for Ravi in the more recent storyline who is a talented mathematician.

Note Added Jan 2008: I am surprised to say that this book received a negative review in the latest issue of the Notices of the AMS. So far, everyone I had heard discussing this book had loved it. I have long believed that for every work of fiction there is someone who will love it and someone else who will hate it. Calegari's review seems to support this conjecture of mine. However, I would still like to point out that among a very large and diverse group of readers, this book is held in very high esteem. Still, Calegari has some valid complaints and you can read about them here.