Eschewing interesting anecdotal tidbits, this short book aims for the heart of the principal controversies over the foundations of mathematics. The reader is given the basic logic and mathematics needed to understand the main points of logicism, Zermelo-Fraenkel set theory, intuitionism, finitism and Godel's incompleteness theorems. The chapters are well-written and lucid. You will doubtless pick up something of value if you merely read the book. You will gain more if you study it and do some of the exercises (which do not come with an answer section).

I found that this book utilized a little too much set theory for beginning students. If the author could have given more concrete examples, perhaps from group theory or simpler ones from analysis or number theory, it would have been much better. For students wanting a more lucid exposition of proof techniques, I highly recommend, "100% Mathematical Proof" by Rowan Garnier and someone else,whos name escapes me at the moment. "100% Mathematical Proof" is far superior to this book, and it has answers to the exercises which is crucial to the beginning student learning on his/her own. Velleman needs to bring the abstract nearer to the concrete for the beginning student.

A mathematically inclined student can expect to reap a bountiful harvest from D.J. Velleman's 'How to Prove It.' You needn't be a computer type to benefit. In fact, the book avoids computer gobbledygook and, in a highly disciplined manner, hones in on the essentials of proof techniques. Though Dr. Velleman's overt aim is to familiarize the student -- no advanced math necessary -- with the reading and writing of mathematical proofs, he also succeeds admirably in teaching basic logic and set theory as a useful mathematical tool, rather than as a mere corpus of interesting ideas. Velleman writes in a spare, lucid style and his exercises are well chosen to illustrate his lessons, though for some reason the book omits the customary answers to alternate exercises, which is useful for someone, such as myself, engaged in self study. Even so, other writers could take pointers from Velleman. I had very little trouble using the book for self-instruction, which is more than I can say for the Schaum's guides and numerous other math textbooks. I found no significant errors in the text or exercises, though Velleman and I did have a bit of an email dustup over 'vacuous truth' .... A quibble: Velleman omits mention of the foundational problems of set theory, other than to stick Russell's paradox in as an exercise. The final (and very good) chapter gives us Cantor's theorem without mentioning Cantor's paradox. Though beginners may shrink from foundational subtleties, a few more words would have been useful. Yet, all in all, this is an elegant, succint and enormously useful text.

The strength of this book is that it tries to develop an algorithmic structure for the approach of proofs that is very similar to computer programming. This means that the logic is easier to understand because of the way he standardizes his symbols and lays out the logical flow of different prove techniques. Many examples are worked out in detail. I recommend this book to anyone (especially engineering students) without formal training in mathematics (but who can program computers), who need to understand very formal mathematical material. The presentation is strengthened by the author's use of basic set theory to illustrate the proof technique. This means that the results you're trying to prove are often pretty obvious, but this allows you to concentrate on the technique of proof in question. Also check out Polya's book of the same name.