Here’s a review of a math text covering a vital area of mathematics that rarely receives a course in itself.
Title: How To Read and Do Proofs – Fourth Edition
Author: Daniel Solow
Original Publication Date: This edition copyright 2005
This deals with various manners and methods of doing rigorous mathematical proofs, with examples.
The very clear, and completely explicit step by step instructions for each style of proof.
Many of the exercises cannot be solved using only the material in the preceding chapter. The author suggests that this text is appropriate for a student right out of high school. Although the body of the text certainly is, large numbers of the exercises depend on higher level mathematics that would not be yet covered.
The entire purpose of this text is to clarify a vital but rarely explained area of mathematics. Thus, the clarity of the text is extremely important. Thankfully, Solow delivers. The style of the text is more like an informal lecture than your traditional math textbook, eliminating the typical need to mentally translate the formal jargon into normal, comprehensible speech. I give it 5 out of 6.
The structure of the text is logical and easy to navigate. The sequence in which the proof techniques are introduced is carefully chosen, progressing naturally from the simplest form through various related techniques until the most complicated techniques are finally dealt with. Charts on the inside covers make it easy to reference later when you want to look up a specific technique. I give it 6 out of 6.
The examples given are all appropriate, and they are dealt with explicitly and methodically, driving the text rather than supplementing it. Because of this efficiency, there aren’t as many used here as there are in other texts. (The material is taught effectively by integrating one or two examples, rather than describing things vaguely before giving six less effective examples.) I give it 5 out of 6.
The exercises are chosen to demonstrate the techniques from that particular chapter. Unfortunately, that means that some of them need math content more advanced than that available to the earliest stage of reader described in the introduction. There are a series of appendices that provide additional information on these topics, but if you’re reading the text from cover to cover (as most people would) that’ll be too late for the first exposure, and irritating to go back and find them the second time around. Hopefully, the fifth edition will take some of these exercises and move them to the appendices, leaving each proof chapter with a list of “further problems” from later in the text to more experienced readers to find them quickly, while allowing the less experienced readers to deal only with those that they can handle at that stage. I give it 3 out of 6.
The text is very complete, including every technique for writing proofs that I’ve ever seen or used. I give it 6 out of 6.
The editing is very well done, as one would hope in a fourth edition. I give it 5 out of 6.
Overall, the pros easily outweigh the cons. I recommend that anyone taking college level math track this text down soon. Those of us who have taken those courses well know that proofs are an unavoidable part of the curriculum, and yet there is rarely explicit instruction given on how to do them. I found my way through the courses primarily by instinct; this text formalizes the structure in a way that makes the process dramatically more efficient. I give it 5 out of 6.
In total, How to Read and Do Proofs: Fourth Edition receives 35 out of 42.