This might turn out to be a boring, shallow book review: I merelyloved the book.
First, what's not to love about the subject matter? (Perhaps not all agree with it — in particular, we might not see a need to replace limits, and we might not be comfortable with, perhaps unnecessarily, losing the Law of the Excluded Middle. Still, it's fun to be free of this law for awhile , knowing we can always find it again if we want to…)
Second, the explanations are so clear, so considerate ; the author must have taught the subject many times, since he anticipates virtually every potential question, concern, and misconception in a student's or reader's mind.
Third, the poet in me is partial to short books, which nowadays seem to be rather rare.
Fourth, the Introduction really and truly gives us a very good sense of the content, and can proves quite helpful to refer back to.
Since this review is entirely complimentary, probably the best thing for me to do here is to offer a selection of enticing passages:
Page 7: "Nonzero infinitesimals can, and will, exist only in a 'potential' sense. Nevertheless, as we shall see, this potential existence will suffice for the development of 'infinitesimal' analysis in smooth worlds." and "Just as the perimeter of a polygon is the sum of its finite discrete collection of edges, so any continuous curve should be representable as the 'sum' of an (infinite) discrete collection of infinitesimally short linear segments — the 'linear infinitesimals' of the curve."
Page 17: "We allow for the possibility that locations may not be presented with sufficient definiteness to enable a decision as to their identity or distinguishability to be made."
Page 23: "[the microneighborhood is] 'large enough' to have a slope but 'too small' to bend."
Page 27: …"every map … has a derivative."
Page. 49: "The 'curvature' of a curve in S is manifested in the microrotaton of its straight microsegment as one moves along it."
Page 85: "in classical complex analysis analyticity is not generally implied by satisfaction of the Cauchy-Riemann equations; one requires also that certain continuity conditions be satisfied. In S, however, these extra conditions are automatically satisfied. . ."
Page 90: "…any R-valued function [on the set of all kth-order infinitesimals] behaves like a polynomial of degree k."
And on page 93 "the three natural microneighborhoods of zero" make their appearance.
Sound tempting? The Exercises are also fun to work on, or to simply glance at. For example, page 25: "Call two points a,b in R neighbors if (a – b) is in [the basic microneighbourhood of zero]. Show that the neighbour relation is reflexive and symmetric but not transitive" and "Show that any map… is continuous in the sense that it sends neighbouring points to neighbouring points."
Also commendably, the book is complete in that it compares this "smoothe world" theory of infinitesimals to Robinson's non-standard analysis, listing the differences. It also has an Appendix which, for those who understand such things, treats everything axiomatically, in particular via category theory. Moreover, as an undergraduate text, the book minus that chapter and perhaps the next-to- the-last chapter would make a wonderful course, one which I'm sure I'd enjoy teaching.
Marion Cohen is Professor of Mathematics at the University of the Sciences in Philadelphia.