Introduction to Mathematical Structures and Proofs

It is pretty much our common lot as mathematics professors to be called to teach, from time to time, the “transition-to-higher-mathematics” course now all but ubiquitous in undergraduate programs across the fruited plane, if not beyond. The idea is to soften — to whatever extent this is possible — the shock mathematics majors and fellow travelers (e.g. computer science majors) suffer when they travel from the computation factories of the first two years of calculus to real analysis, group theory, topology, &c.

The vast majority of our charges (and yes, we are more and more in loco parentis now that eighteen is the new twelve) enter upper division in blissful ignorance of mathematics’ true nature. Indeed, these youths occupy a fool’s paradise in which their smug ignorance is exacerbated by the condition that none of us relish the experience of doing justice to such things as proving that differentiability implies continuity: doing this causes us only to be met with blank stares from our charges, appended by the all-too-frequent “will this be on the test” plea. And so we are more and more resigned to perpetuating the condition that calculus is there for the engineers to use, not for the one or two mathematicians in the group to appreciate. And we pay the price.

Yes, the time comes to pay the piper: the small handful of kids who go on to higher things (often for questionable reasons, to be sure) must learn how to think like mathematicians, not engineers (to pick an obvious target), and therefore we have created the aforementioned “transition” course. It’s clearly a huge cash-cow for publishers, since when one integrates over the nation’s mathematics and computer science majors, one gets a very decent population to hit. And there are accordingly scores and scores of text books aimed at guiding our post-calculus students in the direction of genuine mathematical reasoning, i.e. the business of doing proofs — so many sow’s ears, so few silk purses, but try we must.

And this brings me to the book under review, Larry Gerstein’s Introduction to Mathematical Structures and Proofs. Let me say first off, that given the realities on the ground, i.e. the state of affairs I vented about above, it’s quite a good entry in the given text-book competition. If you’re faced with a dozen or so majors, only one or two of whom actually want to do proofs and struggle day-in day-out with difficult mathematical themes (and really can’t imagine doing otherwise — so I guess I’m talking about one or two students every five or ten years, on the average), you need a solid, easy-to-read text, not paced too fast, with examples galore, and chock-full of exercise sets covering a spectrum of degrees of difficulty. Well, Gerstein does all that.

Additionally, he hits all the topics most of us would vote for as non-negotiable: for example, we encounter baby formal logic, a decent load of set theory (Cantor on almost every page as the middle chapters march on), a solid treatment of equivalence relations, and a proper representation of proof by reductio ad absurdum, material on mathematical induction and recursion. So all the i’s are dotted and the t’s are crossed, and that just leaves the question of what else Gerstein offers by way of mathematical playgrounds wherein the kids can learn the feel of some genuine mathematics. It turns out that his choices pretty much match mine, for what it’s worth, as he presents two large(-ish) chapters on combinatorics and elementary number theory. In the former we encounter a beefy treatment of permutations, and in the latter divisibility properties of the integers get the ball rolling — after a while Gerstein is sure to give some of airplay to the primes: he’s a number-theorist after all.

So there it is: a good book for a course that is notoriously difficult to teach, given how much of a shock it is for our students. One last observation: the print in the book is very large. Why? Well, I guess it’s kinder to the students to go this route than to foist something like the small and dense type of Hardy’s A Course of Pure Mathematics on them…

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