....... a superb source of variants of the problem, paying careful attention to the hidden assumptions behind the problems, written in a witty accessible style. The reader will find discussions of many variants -- progressive versions, Bayesian treatments of the problems, computer simulations, quantum versions, information-theoretic representations, common cognitive fallacies associated with the problem, and much more. This is a model of how to accessibly introduce mathematical material at an elementary level that is not a mere popularization of the material. Although not suited, nor designed, to teach elementary probability theory, it could usefully serve as a supplementary text for the stronger students in such a course, especially in conveying the ideas of replacing an appeal to intuitions by an explicit mathematical treatment and the care needed in setting up under-specified problems for a rigorous mathematical analysis.
(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?
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