Mathematicians, as a whole, are well-versed in the rules of logic. How often do we consciously apply that logic outside of the classroom or our research, though? *The Art of Logic in an Illogical World* challenges us to do so. This book provides instructions to aid both mathematicians and non-mathematicians in building connections between abstract thinking, current issues, and our emotions. Split into three parts, the book covers the power of logic, the limits of logic, and beyond logic. I found that the book also inspires me as a teacher, and I discuss that in the final section of this review.

**The Art of Logic**

In the current political and social climate, we seem to be split into two groups of people: those who argue (loudly) and those who have given up on changing the opinion of the other side. One aim of Cheng’s book is to show us *how* to argue (not *what* to argue). Perhaps the angry discourse so prevalent today would be lessened by simply having more people knowledgeable of how to argue effectively and meaningfully. The necessary definitions, such as converses and negations, are taught to the reader through unusual contexts, helping bridge the distance between logic and daily discussion.

Learning how to argue goes beyond knowing basic logic, although logic is taught in an efficient way in this book. For example, I have known about implications, contrapositives, quantifiers, and so on for many years now. I have not, however, applied these ideas to illuminate statements about issues like healthcare, climate change, or sexism. Cheng does not sidestep controversies such as these, and she is also willing to share personal information, such as her battle to lose weight, in her examples.

We tend to view logic as black and white, or true versus untrue. Since most problems in the world are not binary, there is a flaw in relying on logic. One demonstration of this in the book is through people’s opinions on social services. Should we provide these services despite the chance that some people could receive unwarranted assistance? Should we not provide them because everyone is responsible for themselves? This is not a two-sided issue, as most people could be convinced that there is a gray area. Our problem as a society is remembering that the gray area exists and trying to decide where to draw the line. Gray areas, as with many concepts in the book, are nicely abstracted through mathematical constructs like Venn diagrams and induction. Through abstracting real world situations we can see the similarities or differences between those situations, or simply understand them better. The flaws with abstractions are also discussed, of course.

The last part of Cheng’s book emphasizes the drawbacks of abstract logic and considers the role of our emotions. We are given tools for employing emotional logic, such as using analogies to showcase an issue in a clarifying or touching way. We are also given advice on recognizing emotional illogic, like false equivalences. I had never thought much about the interplay of logic and emotions, but I believe I harbored a subconscious opinion that they should be separate. Instead, we should “find the explanation behind the emotions, uncover the difference between that logic and the one you are trying to convey, and use emotions to help bridge that gap.” I appreciated Cheng’s comparison of a good argument to a well-written proof. It must have correct logic, but also must be written convincingly and effectively.

**Logic in my Teaching**

When I began reading *The Art of Logic in an Illogical World*, I was not expecting to view it as a course textbook. Of course, it is not a textbook in the traditional sense; there are no exercises or index, for instance. As I progressed through the book, however, I often stopped to jot down a page number of a diagram, example, or explanation that I thought would benefit my students or myself as a teacher. Much of the book especially applies to students learning to prove, but a lot of the ideas might be used in a class like liberal arts mathematics.

Two of Cheng’s early explanations struck me, as she was able to clarify ideas that I have struggled to expound to students. Cheng likens doing mathematical proofs to athletes training at a high altitude. When we prove theorems, our minds are exercised to such a degree that it becomes easier to form or follow logical arguments in everyday life. This analogy reminds me of mathematics teachers comparing learning mathematics to practicing sports or a musical instrument, but it takes that comparison a step further. The second instance of a well-formed explanation was about the meaning of “theory” in mathematics. A theory in science is different than, say, the Fundamental Theorem of Calculus in mathematics, which is proved to be true using logic. As long as mathematicians agree on logic, we agree on mathematics. It is better said in the book as “Logic is to mathematics as evidence is to science.”

While some parts of this book described concepts that I already knew or agreed with, albeit in a more successful fashion, other sections challenged me with new ideas to consider. In a chapter about axioms, which I have considered mostly in the context of geometry, Cheng lists her personal axioms: kindness, knowledge, and existence. These three points lay the foundation for the rest of her beliefs. This made me wonder what my personal axioms are (I still have not decided). My husband and I share many beliefs, so do we have the same axioms? Do the people that I disagree with politically have different axioms? Then I thought about whether I have teaching axioms that set the stage for how I run a classroom and how those might conflict with my students’ learning axioms.

There were numerous other sections of the book that were illuminating, even for a pure mathematician like myself. Cheng’s passages on the directionality of logic through broccoli and ice cream, the principle of induction as explained through cookies, and an extension of a factor diagram to explain types of privilege are just a few parts of the book that stood out to me. That last example, concerning privilege, is a good one to show how this book motivates the reader to push at the wall separating abstract mathematics and “real-world” issues. If we can take the prime factors of 30 (2, 3, 5) and replace them with types of privilege (rich, white, male) to clarify the relationships between these privileges, the doors are opened to countless other enlightening comparisons. Seeing abstract mathematics applied to current issues would benefit students, as it can make difficult areas of mathematics more accessible and applicable.

Cheng’s book covers the basics of logic, argumentation, and emotion through humor and an excellent use of illustrations like ice cream, sexism, and Gödel’s Incompleteness Theorem. As long as the reader is not afraid to be confronted with controversial issues and theoretical mathematics (even homotopy equivalences!), this book will be worthwhile.

Mindy Capaldi is an associate professor at Valparaiso University. Her current research area is mathematics education, but she enjoys the Scholarship of Teaching and Learning realm. Her favorite area of mathematics is Abstract Algebra. She is a fan of reading fiction and doing math, and spends much of her time on these two activities.