You are here

Proof Patterns

Mark Joshi
Publication Date: 
Number of Pages: 
[Reviewed by
Jeff Ibbotson
, on

One of the first questions I ask in reviewing a book is “who is the intended audience?” Even after finishing this short book (seemingly) on proof techniques, I found I could not answer that question to my satisfaction.

There is plenty of nice mathematics in the book — Number Theory, Combinatorics, Discrete Geometry, Algebra, Set Theory and Analysis. All of this is, of course, at a fairly elementary level and highlighted by proofs by that involve some cleverness in their construction. In this regard there is much overlap with a course that has become a standard requirement for mathematics majors at most institutions — the “Introduction to Proofs and Formal Math” course that often precedes a first course in Real Analysis.

Indeed, the author states that his own approach to the “transition to proofs” course is to offer a pattern-based approach so that students can learn to recognize the use of certain ideas across many mathematical subjects. This seems to work fine in the early chapters (“Induction”, “Double Counting”, “The Pigeonhole Principle”) but is less useful in the latter ones (“Linear Dependence, Fields and Transcendence” for instance), where the techniques and subject matter are rather closely bound together.

In each of the 20 small chapters are a few problems for practice but not nearly enough for actual use in a Proofs course. I am forced to conclude that the book is best used for outside reading and yet, there are many better presentations of such material for the independent reader (start with Proofs from The Book). So who is this book for? I have no idea.

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.