Misplaced emphasis on tests of significance is indeed arguably one of the greatest "wrong turns" in twentieth century science. This point is widely accepted amongst academics who use statistics, but perversely the innate conservatism of authors and academic journals causes them to continue a bad tradition. All this makes a great topic for a book, which in the hands of an inspired author like Steven Jay Gould might have become highly influential. The book under review is perfectly correct in its central logical points, and I hope it does succeed in having influence, but to my taste it's handicapped by several stylistic features.
(1) The overall combative style rapidly becomes grating.
(2) A little history -- how did this state of affairs arise? -- is reasonable, but this book has too much, with a curious emphasis on the personalities of the individuals involved, which is just distracting in a book about errors in statistical logic.
(3) The authors don't seem to have thought carefully about their target audience. For a nonspecialist audience, a lighter "how to lie with statistics" style would surely work better. For an academic audience, a more focused [logical point/example of misuse/what authors should have done] format would surely be more effective.
(4) Their analysis of the number of papers making logical errors (e.g. confusing statistical significance with real-world importance) is wonderfully convincing that this problem hasn't yet gone away. But on the point "is this just an academic game being played badly, or does it have real world consequences" they assert the latter but merely give scattered examples, which are not completely convincing. If people fudge data in the traditional paradigm then surely they would fudge data in any alternate paradigm; if one researcher concludes an important real effect is "statistically insignificant" just because they didn't collect enough data, then won't another researcher be able to collect more data and thereby get the credit for proving it important? Ironically, they demonstrate the real world effect is non-zero but not how large it is ......
(i) [as conclusion of "what is mathematics/mathematicians?" discussion] A mathematician is someone who sees opportunities for doing mathematics.
(ii) Being a research mathematician is akin to being a writer or an artist; any glamour that's apparent to outsiders fades quickly in the face of [the reality]. Your satisfaction must come from the high you get when you suddenly, for the first time, understand the problem you're working on .....
(iii) As your career develops, the worldwide mathematical community will be increasingly important to you. You will become part of it, and then you will have a home in every city on Earth.
A nice introductory chapter extols the breadth of uses of mathematics, and could be used by Math Dept web sites to recruit majors. There is also a well-balanced "pure or applied" chapter. But the rest of the book identifies "mathematics" as "theorem-proof mathematics", as illustrated by quotes such as
(i) There has been a spate of popular math books in recent years ...... there are even books on the applications of mathematics.
(ii) The true mathematician is not satisfied until the statement is proved.
This attitude is irritating to those of us who regard this identification as akin to identifying "visual art" with oil painting. What matters isn't the tools, it's how competently and creatively you use them.
A final irreverent thought: instead of A Mathematician's Apology, wouldn't The Screwtape Letters have made a more intriguing model?