Some books are of such depth that it is impossible to completely digest all that they contain even after multiple re-readings. Many achieve this through their level of technicality, or through sheer obscurity. The true gems are those that achieve it through clear intelligible discussion of deep concepts. Books like this point outside of themselves, leading one to whole new conceptual worlds. They force new connections to be made in the reader's brain. I reserve my highest recommendation for books of this type, and "The Mathematical Experience" is certainly one of them.
Popular books such as Ivars Peterson's "Mathematical Mystery Tour" and Keith Devlin's "Mathematics: The Science of Patterns" excel at giving the non-mathematician a glimpse into the world of modern mathematics, and an appreciation of the beauty and interest found therein. Depending on the level of sophistication of the reader, some popular math books are more appealing than others, in as much as they convey more or less actual mathematical knowledge. However I would venture to guess that these works hold little interest for real mathematicians, being much too shallow in their description of modern problems, even outside the specialized field of the reader.
Davis and Hersch on the other hand should strike a chord with most practicing professionals, as well as with the lay audience. As the authors state in the introduction, the layman reader may at times "feel like a guest who has been invited to a family dinner. After polite general conversation, the family turns to narrow family concerns, its delights and its worries, and the guest is left up in the air, but fascinated."
We receive the same service of exposure to intriguing mathematical ideas as in other popular books, but we also get healthy doses of philosophy and history. We get glimpses of truly mind-boggling (or mind-expanding ... the authors would perhaps say that bogglification is a primary path to expansion), mathematical concepts such as the Frechet ultrafilter, the truly huge integer known as a moser, or Weiss's restatement of the Chinese Remainder Theorem which is so abstract and generalized as to defy the understanding of all but a handful of practicing mathematicians.
The book tackles problems of mathematical experience which are tough because they fall into the realm of philosophy: the meaning of proof, the goal of abstraction and generalization, the existence of mathematical objects and structures, and the necessary interplay between natural and formal language, or between algorithmic and dialectic processes. What is amazing is that Davis and Hersch make these ideas not only accessible to an intelligent layman, but also interesting and vital, without (I presume), losing the interest of real mathematicians.
Rather than a zoo of mathematical curiosities, the book is an anthology of essays about the practice of mathematics, with illustrations ranging from the elementary to the extraordinarily deep. I suspect that the questions "What is mathematics?" and "What does a mathematician actually do?" are rather off-putting to the majority of professionals in the field. But "The Mathematical Experience" asks these questions, and rather than giving a terse answer, takes them very seriously and fearlessly analyses them from a variety of stances. Of course, the authors don't presume to give definitive answers. They do, however, provide much food for thought ... so much that the reader is likely to come away from the book transformed.
If you are not a mathematician, but a curious layman, "The Mathematical Experience" is the best place to go after you've read William Dunham, Ivars Peterson, Keith Devlin, Ian Stewart, or others like them. If you are a student of mathematics, or a student or practitioner of any other science, you'll do yourself a great favor by reading this book. If you are a mathematician who generally dislikes and avoids pop-math writing, give this a try. You may be very pleasantly surprised.