Customer Review

February 14, 2010
This is on the whole a pretty decent book "intended primarily for liberal arts students" (p. vii). There are some occasional intriguing remarks and stimulating exercises here and there that make this book slightly more interesting than the traditional ones for this audience. But that is not saying much, of course.

I shall argue, however, that the book does its readers generally and its intended clientele especially a major disservice by placing undue emphasis on modern so-called rigour. Although the author claims that his "approach is basically historical in nature" (p. ix), he shares the contempt for early calculus so fashionable with his colleagues, propagating such myths as the ridiculous claim that "mathematicians of the Enlightenment were using their instinct more than their intellect" (p. 310).

Another very harmful myth is this: "What does it mean to assert that the area pictured above is 6 square units? ... Is it nonsense to speak of the 'area' inside a curved figure? This question was profoundly considered long ago by Archimedes" (p. 238).

Riemann sums supposedly "lead to a clear understanding of what is meant by the area beneath a curve" (p. 255), but despite being allegedly so "clear" we can't actually do much with this sterile notion. In fact we cannot even prove that areas as given by this "clear" definition even exist: "the proof of the existence theorem is better left to a course in analysis" (p. 264). Similarly, discussions of both parts of the fundamental theorem of calculus end on the same disappointing note that a complete proof is "better deferred to a course in analysis" (p. 243, 267).

The intended readers---liberal arts students, remember---will of course not go on to this course in analysis. This will most likely be the last they see of mathematics. They will be left with the impression that a central issue of the calculus, on which mathematicians have spent thousands of years of "profound" thought, is whether the notion of area of a curved figure is "nonsense." They will also be left with the impression that they ought not think for themselves but should rather "defer" understanding of key theorems to the experts, their own feeble attempts being restricted to mere "instinct" rather than "intellect."
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