As students make the transition from the calculus track to abstract mathematics, should they be given a thorough introduction to the concepts, definitions, and theorems of the new fields which they'll be studying in the coming semesters, or should they first deal with the process of higher mathematics, conjecturing statements and then attempting to prove or disprove them? Should they do a run-through of the ``big ideas'' or should they practice dealing with just a few of them? Those that lean towards the survey approach will find a lot to like in A Primer of Abstract Mathematics by Robert Ash, a very elegant treatment of a sizable sampling of topics from the abstract side of mathematics. However, those who feel that process is more important than content will have the same problem that Ash himself seems to have with this book. What do we do with it?
Ash has selected his topics with the stated goal of preparing the reader for abstract algebra. The book has chapters on logic, counting, elementary number theory (including a section on algebraic structures), set theory, linear algebra and linear operators. The set theory chapter is highly informal, an approach the author credits to seeing Munkres' exposition of the Maximum Principle in his topology text. The chapter on linear operators begins with an introduction of the Jordan Canonical Form, which is then used to simplify the development of minimal and characteristic polynomials, adjoints, and normal operators. Each chapter has roughly half a dozen sections, each of which ends with a handful of problems. Solutions, or at least sketches of them, are given in the back for every problem. Proofs are kept short, and in keeping with the intention of making the book accessible to readers with limited experience in abstract mathematics, are often preceded by an illustrative and motivating example.
The strength of this book is the writing itself. Ideas are developed in a conversational (but not chatty) style. The proofs are terse and compact, but remain complete and readable, striking an impressive balance between informality and rigor. Ash is careful to point out important issues which may lurk behind the smooth exposition; for example, flagging the issue of well-defined operations when dealing with residue classes, or pointing out that an example, however "typical", is not a proof. (This doesn't prevent him from using one, however, in the case of the statement that there exist integers a and b such that ax + by = gcd(x, y), arguing that the lack of rigor is a small price to pay for the increased clarity of the exposition). Anyone who deals with the struggle to find the balance between including too much and too little when composing a mathematical argument will find many good models here.
But the question remains: what do we do with it?
The author suggests in the preface that his book might be used as a text for a course in mathematical thinking, and the publisher advertises in its catalog, "Ideal for a course that serves as a transition to advanced mathematics." As a principal text for a transition course, though, this book has some significant weaknesses. If the intent of such a course is to introduce students to the concepts of higher mathematics, they're forced to deal with them very quickly. Each chapter is only about 20 pages long , (chapter 5 on linear algebra is a bit longer), allowing only about three pages per section. For example, groups, rings, integral domains, and fields are defined and examples of them are given on pages 51-54, and homework problems 3 and 4 on page 55 ask the reader to explore the polynomial ring F[x], ultimately deciding what needs to be done to make it into a field. When abstract vector spaces, linear independence, bases, and the idea of without loss of generality are introduced in the space of just four pages, things are coming too fast.
The survey approach ignores an important part of the transition from computational mathematics. What makes higher mathematics different is as much procedural as it is conceptual; the number and complexity of new concepts should take a back seat to the practice of conjecture and proof. Other textbooks (Kenneth Rosen's Discrete Mathematics and its Applications, for example) appropriately focus on fewer topics, recognizing that students need time to develop some facility with new ideas before they are able to construct arguments concerning them. Other approaches focus on a single topic, encouraging the students to pursue questions on their own, demonstrating to themselves the need for proof as opposed to substantial but anecdotal evidence.
Ash suggests that this text might be appropriate for motivated students who may not have much mathematical experience but want to have a sense of what lies beyond calculus. He also suggests that it might be used as a resource for graduate students or professionals in applied fields who need some experience with abstraction. But, for each audience, there are better choices. Those students without mathematical experience looking for a survey of the subject now have several excellent books to choose from intended specifically for them. Readers from applied fields who desire or need to understand the connected abstract ideas are better served by other, more complete sources. A chemist who wants to know the group theory behind the symmetry properties of molecular vibrations, for example, will learn much more from an abstract algebra text than from the three pages given here.
Finally, the author presents this book as a model of abstract reasoning written for an audience with limited experience in abstraction. In this respect, it excels, but readers with more experience than Ash envisions will reap greater benefits. It's asking a lot for students who are just being introduced to the notion of mathematical proof to appreciate the impressive balance Ash is able to strike in his writing, but a reader who has already struggled to find that balance will appreciate it immediately.
Reading this book brought back memories of my first experience with Nathan Jacobson's Basic Algebra I. It was very difficult to learn from; the term "characteristic" wasn't even in the index, for cryin' out loud. But now, after having worked through the material (with some help from Hungerford's book), I can read Jacobson and appreciate what an exceptional book it is. Learning from Ash's book would be a similarly frustrating experience, but reading it after having dealt with the topics one at a time, and in depth, is a much more satisfying and worthwhile experience.
So, what do we do with this book? Here's my suggestion. Students who do not go on to graduate school will most likely not have time to pursue interests in abstract mathematics. This book might serve as a souvenir of their studies, a memoir of those subjects and theorems from their undergraduate years. It is an elegant compilation of the main ideas from their courses in abstract mathematics.
Steve Morics is assistant professor of mathematics at the University of Redlands. Having never before lived west of the Mississippi, he's trying to adjust to life in southern California, with only two seasons (dry and wet) instead of four, freeway interchanges that look like modern sculptures, and summer temperatures which can range from the 110's during the day to the low 60's at night. His first mathematical love was combinatorics, and he's now working on research in mathematics education and the connections between mathematics and politics.
Appendix: An Application of Linear Algebra.