Says Kane on p. ix:

This is not your traditional transitional textbook. The goal of this book is to give the student precise training in the writing of proofs by explaining what elements make up a correct proof, by teaching how to construct an acceptable proof, by explaining what the student should be thinking about when trying to write a proof, and by warning about pitfalls that result in incorrect proofs.

That says it all, of course: this is what we — or at least I — try to do when teaching our “transitional course.” It really all entails the essentials and even accidentals pertaining to an apprenticeship regarding the ultimately ineffable business of proving theorems. After all, it’s an art: it’s like teaching someone who has some talent (oh, would that it were so, now that we are called to teach so many hostile fellow-travelers how to “do” mathematics, if only at a beginner’s level) how to sketch and then fill out a picture or a musical composition, except for the nasty added element that mathematical art is supposed to be true and correct by absolutely objective standards. Well, leaving aside the hardest part of this job, i.e. the question of how to have ideas or even how to “see” mathematical structure (for lack of a better word), we are accordingly called to cover a repertoire of structured exercises in the broad sense, i.e. to develop certain themes with the students, resulting in their acquisition of corresponding skills (an unfortunate word), eventually adding up to some sort of mathematical sophistication and a particular way of thinking. Granting that premise let’s consider how Kane goes about this.

He starts with an essay on what proofs are and why we write them, setting the stage for a long chapter titled “The Basics of Proofs.” This chapter contains a lot of what a typical transitional course contains, e.g., proofs about sets, about even and odd integers, the reals (including the triangle inequality: it’s *de rigeur*, after all, across many spectra), and then stuff on functions including the business of domains, codomains, injectivity, surjectivity, and composition. It is noteworthy that Kane somewhat audaciously includes a section dealing with “templates for proofs.” In this regard his book is already distinct from the herd: I guess it’s *Realpolitik* in that if we’re going to try to get the youths to mimic us we should give them something to mimic.

Beyond that, this chapter indeed develops a goodly chunk of the basics. The complementary material in a standard transitional course includes such things as induction (both flavors), the yoga of equivalence relations, sundry themes in elementary number theory, and the baby-theory of infinite sets, to name a number of popular favorites. And it is here that Kane departs form the script in a dramatic way: he immediately turns his attention to analysis. Granting another premise, namely, that undergraduate analysis is indeed a very good place for a rookie to cut his teeth, what follows is really quite excellent.

To wit, the book’s remaining chapters are a curriculum for a standard undergraduate analysis course, *viz*. we get, in sequence: limits, continuity, derivatives, Riemann integrals, infinite series, sequences of functions, the topology of \(\mathbb{R}\) and metric spaces. I guess it’s fair to say that one could use this material for a two-semester or (better?) three-quarter sequence taking a kid who’s done very well in calculus straight into analysis, modulo an interval spent on basics, as per Kane’s first chapter. And make no mistake, the material Kane ends up covering is pretty sporty. For example, he gets all the way to the contraction mapping theorem (cf. p. 328), which is proven in wonderful detail, in a format Kane employs throughout the book: the proof proper (as distinct from the surrounding discussion, which, by the way, is quite ramified and pedagogically sound) is fitted into a light-green colored box where it is laid out using bullet points. It is therefore extremely explicit and easy-to-follow, at least in principle. Indeed, I am reminded of the by now Jurassic practice of writing out a geometry proof in a “T format,” with statements and reasons: that certainly resonated with me back in the days of the Lyndon Johnson administration — well, I was in the Netherlands at the time, so I guess I should use a different measure; how about when Brazil was between World Cups no. 2 and no. 3? (Ah, yes, I remember it well…)

So, to sum up, I think this is indeed a fabulous book for the kind of course I just suggested. I think that it will indeed serve as Kane projects it should, and the surviving student will truly know a good deal about writing a mathematical proof, in fact, about thinking about the problems and assertions beforehand and then going about the task of constructing the proof. Still, some kids will fail, but that’s unavoidable: you can’t teach even a basic tune to a tone-deaf subject (or victim). But I think that Kane’s approach will indeed make serious inroads in the direction of training our fledgling theorem provers more effectively. Down the line they’ll have to learn about weak and strong induction, the binomial theorem, the Little Theorem of Fermat, different sizes of infinity, and the Cantor-Schröder-Bernstein theorem, but that’s all right. It’s easier to teach a third and a fourth language than it is to teach a second one.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.