How to read and do proofs. An introduction to mathematical thought processes (Q2844619)

The textbook aims to serve as an introduction to reading and writing proofs. Its audience appears to be students who are very weak in deductive and abstract reasoning. (In the reviewer's experience as a teacher, this is the vast majority of undergraduates, at least at public universities in the United States, including the reviewer himself, two and a half decades ago.) Aside from techniques such as construction, specialization, either/or, max/min, and the usual culprits of induction, contrapositive, and contradiction, this sixth edition adds material on more general mathematical creativity, namely generalization, creating definitions, and working with axiomatic systems. It also adds videos at the author's website, which go over the proof techniques.NEWLINENEWLINEThis book does not come across as the usual proof-writing textbook, such as one might find in a course on discrete mathematics or on foundations. The chapters are very short; most are a handful of pages, followed by several pages of exercises. The instructional material is to the point, with well-considered examples and asides on common mistakes. Good examples of the author's thoughtfulness appear in the discourses on pp. 5--6 of identifying the hypothesis and conclusion when they are not obvious, on pp. 28--29 regarding overlapping notation, and on pp. 190--191 of the advantages and disadvantages of generalization.NEWLINENEWLINEThere do seem to be some oversights in editing. Figures 10.1--10.3 on pp. 117--118 contain an error; they state that, for a proof by contrapositive, one both assumes a statement (A), and concludes the statement is false (NOT A). Aside from personal experience, the text itself seems to contradict this (``you work forward from only one statement (namely, NOT B)''), but an instructor who uses this text may wish to point it out to the students. Other issues exist, but are less confusing.NEWLINENEWLINEThe author may consider including in a future edition a technique I call the ``maybe it's wrong'' method, which certainly arises in research: when one is trying to prove an idea, becomes stuck, and doesn't see the way to proceed, try to build a counterexample. Either success or failure can be illuminating.