Here is Bill Gates' review of the book.
Here are some of my thoughts on the book.
Let me begin by saying the book offers a lot of interesting tidbits. Here's one where Ellenberg talks about how mathematicians try to prove a statement by day and look for counterexamples by night. The section where he talks about it is so good that I feel justified in quoting at length:
In fact, it’s a common piece of folk advice—I know I heard it from my Ph.D. advisor, and presumably he from his, etc.—that when you’re working hard on a theorem you should try to prove it by day and disprove it by night. (The precise frequency of the toggle isn’t critical; it’s said of the topologist R. H. Bing that his habit was to split each month between two weeks trying to prove the Poincaré Conjecture and two weeks trying to find a counterexample.)This passage speaks to me for the following reason. Lately, I've been learning about investing in individual stocks (don't worry though, most of my monies is nicely stashed in index funds). After being in the game for 4 months or so, I see that most stock analysts (and investors) are overly optimistic. (There are some structural reasons for this.) Thus it might be a great idea to defend the bullish case for a particular stock (i.e. defending that the underlying business will do well and price will rise) by day and do the opposite, defend the bearish case by night.
Why work at such cross-purposes? There are two good reasons. The first is that you might, after all, be wrong; if the statement you think is true is really false, all your effort to prove it is doomed to be useless. Disproving by night is a kind of hedge against that gigantic waste.
But there’s a deeper reason. If something is true and you try to disprove it, you will fail. We are trained to think of failure as bad, but it’s not all bad. You can learn from failure. You try to disprove the statement one way, and you hit a wall. You try another way, and you hit another wall. Each night you try, each night you fail, each night a new wall, and if you are lucky, those walls start to come together into a structure, and that structure is the structure of the proof of the theorem. For if you have really understood what’s keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true. This is what happened to Bolyai, who bucked his father’s well-meaning advice and tried, like so many before him, to prove that the parallel postulate followed from Euclid’s other axioms. Like all the others, he failed. But unlike the others, he was able to understand the shape of his failure. What was blocking all his attempts to prove that there was no geometry without the parallel postulate was the existence of just such a geometry! And with each failed attempt he learned more about the features of the thing he didn’t think existed, getting to know it more and more intimately, until the moment when he realized it was really there.
In fact, as does Ellenberg, I invite the reader to take one of his/her cherished beliefs and play the devil's advocate by defending the said belief's negation.
Besides shining tidbits such as the quoted passage (see also my earlier post on p-values; there are frankly many), I think that the book doesn't do a good job of what it's purporting to do, that of explaining non-trivial but easy-to-understand mathematics to an intelligent but not necessarily numerate reader. I had prior exposure to most of the mathematical content in the book, so I could just ho-hum my way through bits where the author gets quasi-technical. Had I not known the subjects in greater detail, I would have been worried about receiving such summary treatment - and the suspected dilution of cognitive content. For instance, in one place, the author talks about projective geometry. I am only a little familiar with projective geometry, so I didn't really understand the points of the author. I didn't worry about missing author's points because if I really wanted to understand projective geometry, I would have to start with a more technical text.
I would recommend this book to readers who already know quite a bit of mathematics. For them, it might be an entertaining review of many mathematical ideas' distilled essence. For readers who would like to become more sophisticated mathematically, I would recommend first-order mathematics books instead.