Proofs are the backbone of mathematics. *A Gentle Introduction to the Art of Mathematics* (GIAM) contains chapters on logic, set theory, relations and functions, combinatorics, and cardinality, interwoven with chapters on proofs (direct, indirect, by induction, by exhaustion and cases, etc.). A good choice for “introduction to proof” courses, this book can be used as a classroom textbook, or for independent study. It would also make a great supplement in an introductory course on mathematical reasoning and problem solving.

The book assists the reader to develop facility and proficiency with reading, understanding, and writing mathematical proofs. Each topic is written in an informal conversational style, clearly and in detail. Numerous exercises interspersed in the text encourage readers to test their understanding of new concepts/techniques as they are presented. Additionally, after each section in the chapter there is a set of exercises that are meant to expand the student’s mathematical thinking.

GIAM introduces the reader to the process of mathematical discovery. The author invites the reader to “monkey around” (his words) with the problem until eventually patterns emerge, conjectures become apparent, and possible proof techniques appear. He stresses the importance of multiple exposures and repetitions of the concepts and introducing them in an intuitive manner. In this book the mathematics is seen as a very alive subject that should not scare the student with its symbols or rigorous proofs. Moreover, the author asks the reader to read actively, study the tables, and do the exercises (for some of which hints are provided).

A wide and varied selection of concepts is presented in the nine chapters of the book, including proof techniques and advice on writing particular types of proof. The author encourages the reader to think visually and intuitively. For example, he illustrates Boolean expressions by digital logic circuit diagrams. The last chapter contains three remarkable proofs (two of which are by John H. Conway) that are exceptionally elegant, succinct, and full of ingenuity. Using visual representation and what the author refers as “magic” (clever and creative argument), these three proofs will invite the readers to appreciate the beauty of mathematics as well as to become involved in the art of creating it.

Emphasis is also placed on precise definitions. The author provides imprecise/intuitive definitions (for example, of rational numbers) and later guides the reader through the process of refining that definition. The purpose of this is to make the student comfortable with the mathematical terminology used.

Moreover, the book includes standard subjects of mathematical culture, such as the Twin Prime and Goldbach Conjectures, Russell’s Paradox, the Pigeonhole Principle, the Peano axioms, etc.

GIAM teaches that the art of writing proofs is not just about deductive reasoning, but also includes creative aspects and thinking both visually and intuitively. This is a textbook written to provide the type of mathematical knowledge and experience that students will need for becoming proficient in writing and reading proofs.

A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.

## Comments

## Joe Fields

It should be noted that the paperback version from lulu.com costs $27.30. However, the book is entirely free as a downloadable pdf from the author's website www.southernct.edu/~fields/GIAM.

Furthermore, GIAM is free in the broader sense - it is licensed under the GFDL and the LaTeX source code for the book is publicly available.