Although it does explore fallacies and paradoxes, this book isn’t an anthology of them. As he states in the preface, Bunch strives to weave the fallacies and paradoxes he has chosen to discuss together so that “there is a definite progression from the first chapter to the eighth.” He mostly succeeds, including material ranging from algebraic fallacies caused by division by zero and the danger of overly-relying on figures in geometry to infinite cardinalities to Gödel’s Incompleteness Theorem to relativity.
The first half of the book functions as a “warm-up” for the second half. Bunch assumes that the reader knows high school algebra and a little high school geometry, but uses the fallacies and paradoxes in the first three chapters to introduce more advanced concepts, which he then draws on later in the book. For example, he introduces proof by contradiction as part of a discussion of the danger of relying too much on the figures in Euclidean geometry.
Bunch also uses the earlier chapters to get the reader used to the ways of thinking that will be needed for the later material. One way he does this is by inserting a box with the question “Can You Find the Flaw?” and a hint for how to think about it after presenting the set-up for each fallacy, in order to encourage the reader to engage more deeply with the ideas. Additionally, he introduces ideas that he will develop much more deeply later on. For example, the second chapter both develops sequences and series and uses them to start the reader thinking about infinity, and the fourth chapter examines paradoxes in natural language, such as the Liar Paradox, on which he will build with connected mathematical ideas in later chapters.
Each chapter in the second half of the book has a different theme. The fifth chapter builds on the concepts introduced in the early chapters to explore infinite cardinalities, transfinite ordinals and some of their apparent paradoxes. For example, after showing that the natural numbers have the same cardinality as many of their subsets and as the rationals, he returns to the idea of proof by contradiction in order to use Cantor’s diagonal method to show that the unit interval must have a larger cardinality than the natural numbers. Bunch also discusses the continuum hypothesis, introduces axiomatic systems, and uses some of the language paradoxes from Chapter 4 to introduce the difficulties that self-referential statements can cause in mathematics. Chapter 6, on Gödel’s Incompleteness Theorem and decision problems, builds on the ideas from Chapter 5 in order to give an intentionally simplified explanation of the incompleteness theorem. Bunch also explores undecidability using several examples, including the continuum hypothesis.
The discussion of axiomatic systems continues in Chapter 7, where Bunch introduces models of an axiomatic system and explains how they can have very paradoxical properties. He also explores paradoxes in measurement (e.g. the Banach-Tarski paradox) that result from accepting the Axiom of Choice. Paradoxes in measurement also serve as a lead-in to the final chapter, which is on Zeno’s paradoxes.
Overall, I found Mathematical Fallacies and Paradoxes to be well written and engaging, although there are a handful of jokes that the contemporary reader might find to be in poor taste. The later chapters tackle a lot of difficult concepts, but Bunch does a good job simplifying them for the general reader as well as building up to them in the first half of the book.
For the most part, the ideas in each chapter seem to develop fairly naturally from earlier chapters, but I did find the final chapter on Zeno’s paradoxes less well-connected than the others. There’s also a nice progression in the type of difficulties he examines, beginning with fallacies, moving to paradoxes that we eliminated from mathematics by careful definitions, then to paradoxes in mathematics that we just have to live with, and ending with paradoxes whose philosophical implications are unresolved.
I think this would be a hard book for the typical high school student, but math majors and minors and motivated high school students could enjoy this look at material, much of which isn’t often covered in the undergraduate curriculum. There are passages that I can see using in class or in our math club.
Megan Patnott is an Assistant Professor of Mathematics at Regis University in Denver, CO. Her training is in algebraic geometry and commutative algebra.