Logic is fundamental to mathematics, but mathematicians sometimes find that correct logic is insufficient to win real-world arguments. Following in the tradition of George Boole, Lewis Carroll and John Venn, who provided algebraic or diagrammatic aids to logical calculation, Eugenia Cheng enhances her reputation as a popular mathematics writer with this perceptive analysis of logic and its limitations.

She shows us the dangers of false dichotomies (the existence of white privilege is not refuted by one black person being better off than some white people) and false equivalences (¡°black lives matter¡± does not mean that some lives do not matter), and how arguments cannot be resolved when the two sides rely on different unstated assumptions. In her examples, Cheng bravely chooses sensitive topics ¨C white privilege, sexual harassment, fat shaming ¨C and her lucid exposition illuminates the ways in which misunderstandings create conflict. Although she does not hide her own views, she is not writing about what is right or wrong, but about how to make judgements. We are shown how, rather than ascribing a refusal to be convinced by one¡¯s logical argument to an opponent¡¯s blind irrationality, one can examine the assumptions on which their position is based and thus argue in a way that might actually change their mind.

As the book reaches its conclusion, Cheng writes about emotion and its relation to logic: they are not opposites, but two important aspects of being human that can work effectively together. She describes the roles of emotion and logical rigour in pure mathematics ¨C emotion, important in the initial stages of an investigation, when one is looking for ideas, then takes a back seat to logic when the details of a proof are worked out, and again becomes important when presenting one¡¯s work to others. For those who think that they value logic above all else, there is useful advice on understanding the importance of emotion and the limits of pure logic when it comes to persuading others.

For those who think more visually than I do, a great strength of the book will be its use of diagrams (that perhaps reflect the author¡¯s background in category theory). The tone is generally friendly, if occasionally perhaps a little preachy, and Cheng writes sensitively about delicate topics (although one example, about the logic of staying put in a fire, should have been changed after the Grenfell tragedy). However, I feel that she follows too much her own advice, that one can avoid being wrong by making a practice of qualifying statements with phrases such as ¡°I think that¡±. In doing so, she inadvertently shows how such phrases make arguments less convincing!

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While I sometimes wanted to quibble ¨C for example, I found the analysis of Zeno¡¯s and Carroll¡¯s paradoxes rather simplistic ¨C I was engaged throughout. Overall, Cheng is successful not only in helping readers think more clearly, but in helping them understand why others sometimes appear to be illogical. This book has the potential to help understanding and avoid confrontational arguments that serve only to entrench opposing views. While hardly the ¡°survival guide for our post-truth world¡± promised by the back-cover blurb, the reader will indeed be helped to ¡°see, argue and think better¡±.

Tony Mann is director of the Greenwich Maths Centre at the University of Greenwich.

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The Art of Logic: How to Make Sense in a World that Doesn¡¯t
By Eugenia Cheng
Profile, 320pp, ?14.99
ISBN 9781788160384
Published 5 July 2018