How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (review)

Reviewed by: How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics Barbara Lee Keyfitz (bio) William Byers. How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics. Princeton University Press. viii, 416. US$35.00 When I opened this book, I wondered what it might have in common with Jerome Groopman’s polemical How Doctors Think (2007). Groopman is critical of the education of doctors: he is angry that they do not listen to [End Page 141] their patients or admit their uncertainties, and is upset at the toll in morbidity and mortality caused by their attitude. Like Groopman, Byers is a man with a mission: he wants to correct what he claims are common attitudes about mathematical creativity, and to explicate a new paradigm. And although he does not criticize the education of mathematicians, his book has important things to say about education. The book’s subtitle conveys the message: new mathematics results from resolving ambiguities in mathematical (or physical) concepts, overcoming apparently contradictory evidence, and explaining paradoxical situations. Simply put, mathematicians make progress by clarifying things that had been unclear. Significant results do not follow from simply combining logical arguments, and new mathematics cannot be mechanically generated. Putting all known mathematics into a computer and letting it churn would be as silly as expecting those million monkeys to generate Shakespeare’s sonnets. Who needs to be persuaded of this? Mathematicians, whether they think about thinking or not, know that research is a creative act, though many might feel that Byers undervalues the importance of mastering proofs. And in fact, the duality Byers proposes (algorithm/creation, logic/intuition, rote/ideas) is less clear than he claims. For example, Byers refers to the 1977 proof by Kenneth Appel and Wolfgang Haaken of the celebrated four-colour theorem, but ignores the intense controversy that arose from their use of computers in the proof. Hard cases may make bad law, but I would have liked to see Byers deal with Thomas Hales’s 1998 proof of the Kepler conjecture, which states that the familiar ‘cannonball’ arrangement of identical spheres is an optimal packing. The prestigious Annals of Mathematics, unable after much agonizing to decide whether the proof (250 pages of text and three gigabytes of computer code) was correct or not, accepted the paper with an unprecedented warning. In short, research mathematics is currently experiencing intense conflict on the nature of mathematical proof, and on questions of truth and value in mathematics, at a level that is both deeper and more practical than Byers’s ruminations on the numinousness of defining infinity. The book is intended for non-mathematicians, and although people unfamiliar with undergraduate mathematics and physics will miss the point of some examples (those involving quantum mechanics, for instance), most of the discussion is motivated by very simple examples. Byers is particularly strong on reminding the world that mathematical ideas that are now taken for granted, like continuity and differential calculus, were not only revolutionary but required a long period of gestation. The narrative is peppered with anecdotes from the autobiographies of esteemed mathematicians. Repeatedly, mathematicians recount the pivotal moments when they developed for themselves examples of ambiguity. Fields medallist William Thurston recalls his childhood discovery that the [End Page 142] expression 134/29 was both an instruction (‘carry out this division’) and a number. The tension between the interpretation of a fraction as a process and as a meaningful expression is an elementary but lovely example of the ambiguity that pervades mathematics. To anyone who has experienced such a flash of transcendence, or who has listened to children forming their own pictures of mathematics, this is a key insight into how mathematics is learned. Thus, mathematics teachers should learn to think like mathematicians (even if they think about relatively elementary mathematics), so that they will respect their pupils’ forays into mathematical thought. Byers’s perspective is sunny, his mood enthusiastic. He appears to like his subjects (other mathematicians like himself), and, in contrast to Groopman, he has no quarrel with the way they conduct their business. But, in light of current concerns over the learning of mathematics, one can deduce from his book a message similar...

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