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Bridge to Abstract Mathematics

Ralph W. Oberste-Vorth, Aristides Mouzakitis, and Bonita A. Lawrence
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Textbook Series
[Reviewed by
Allen Stenger
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This book is the newest entry in a growing array of bridge courses, intended to help the student transition from computationally-oriented courses in calculus and linear algebra to proof-oriented courses in more abstract mathematics. The “bridge” part is relatively brief. Most of the book is formalist, abstract mathematics without much guidance on proofs, but is broken into bite-sized pieces and is intended as practice of the techniques learned in the bridge. This makes the exposition choppy, and the index is relatively weak, so the book works better as a workbook than as a text or reference.

The first 50 pages are the bridge: an introduction to axioms, definitions, and proof techniques. Very Good Feature: many examples of fallacious proofs. These are labelled with the neologism “poof” (and the faulty propositions are “absurdums”). The narrative also emphasizes the role of undefined terms (these are labelled “undefinitions”). This section is rudimentary and does not cover much on proof strategy, but seems to be the right level for an introduction. There are good books on proof techniques, such as Velleman’s How to Prove It: A Structured Approach and Solow’s How to Read and Do Proofs, but these are aimed higher than the present work.

One weak area in the present book is motivation: There’s not much explanation of why proofs are important or why they are the central feature of advanced mathematics. The book makes many references to the history of mathematics, but for some reason does not talk about the gradual increase in rigor over the years.

The remaining 150 pages of the book start with the mathematical development of set theory (along with functions and relations), give a thorough and clear development of the complex numbers starting from the Peano axioms for the positive integers and giving two developments of the reals (Dedekind cuts and Cauchy sequences), and give a skimpy introduction to the new subject of time scale calculus. Time scales are an abstraction intended to give a unified treatment of derivatives and differences and of integrals and sums, so that (for example) differential and difference equations can be studied by a common method.

The big weakness of the present book is that it does not go very deep, which may give the student the false impression that there is nothing very interesting in advanced mathematics. Most of the material follows the pattern of “define a few terms, create a few axioms using those terms, prove a few toy theorems, and move on.” In the book’s defense, there are also a large number of supplementary exercises that do have some real mathematical content, but because they are packaged as supplementary and are outside the main narrative, and have no hints or answers, they are likely to be skipped in most courses.

The book is also weak on follow-through. For example, the question of the existence of the square root of 2 is used in several places to motivate the development of the real numbers. But completeness (the least upper bound property) gets short shrift, being an exercise, and there is no discussion of its importance consequences, such as the theorem that an increasing bounded sequence has a limit. Another example is time scales, for which no applications are given. The book mentions differential and difference equations in one sentence and otherwise does not discuss them, so the material is unmotivated and to the novice would appear to be abstraction for its own sake.

A comparable book is Beck & Geoghegan’s The Art of Proof: Basic Training for Deeper Mathematics. It also takes an axiomatic approach, is about the same size, is somewhat less expensive, and covers roughly the same territory. But it exposes a lot more real (and interesting) mathematics and does a better job of explaining why proof is important. It spreads the material on proof throughout the book, bringing it up when it is important, which I think is a stronger approach, as well as reflecting the gradual growth in rigor over the centuries.

Another good alternative, although less comparable, is the budget-priced Dover reprint of Rotman’s Journey Into Mathematics: An Introduction to Proofs. This is aimed lower mathematically, at bright high-school students and lower-division undergraduates, and is not very axiomatic. Instead it builds on the knowledge students should already have of algebra and geometry to attack more difficult problems, including proofs. Like Beck & Geoghegan, this mirrors the historical development of mathematics and the increase in rigor.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

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