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

  • William Byers
Princeton University Press: 2007. 425 pp. $35 9780801885877 | ISBN: 978-0-8018-8587-7

Mathematics is a corpus of immutable, rigorously established facts, right? Wrong. Three recent books argue that mathematics is a profoundly human enterprise, best described as what mathematicians do.

In the most ambitious, accessible and provocative of the three, How Mathematicians Think, William Byers argues that the core ingredients of mathematics are not numbers, structure, patterns or proofs, but ideas. Numbers are ideas, and so are functions and even logic. Ideas are organizing principles. They are neither right nor wrong.

The polymath Arthur Koestler famously characterized ambiguity as involving a single situation or idea that is perceived in two self-consistent but mutually incompatible frames of reference. Mathematics thrives on ideas that harness ambiguity. Byers maintains that great ideas, like Simone Weil's notion of 'true good', have conditions that are contradictory and so seem impossible — they encapsulate ambiguity that is so great as to seem paradoxical.

There's ambiguity everywhere. Zero, for instance, captures both the notion of nothingness, or absence, and the idea that nothing is something. 89/17 signifies both a number and the operation of dividing 89 by 17. Algebraic expressions such as 3x+2 represent a process (take a number, multiply it by 3 and add 2) and a set of numbers. The variable x in an equation simultaneously represents every number and a specific one, and the equality sign also embodies ambiguity. Ideas such as infinity, limits, derivatives and real numbers mediate between seemingly incompatible frames of reference. Their inherent ambiguities require a creative effort to master but give the ideas their power.

Great ideas often emerge in times of intellectual crisis, when ambiguity and apparent impossibility challenge the human impulse to understanding. In Byers' view, the formalist programme to reduce mathematics to formal axioms and their consequences was a great idea that responded to the seeming impossibility of non-euclidean geometries. But neither formalism nor any other great idea comes close to exhausting mathematics. “The content of mathematics,” writes Byers, “cannot be definitively separated from how mathematics is created and understood.”

Credit: A. MARTIN

Byers' view springs from the various facets of his career as a researcher and administrator (and, he says, his interest in Zen Buddhism). But it is his experience as a teacher that gives the book some of its extraordinary salience and authority. He points out that the concepts that cause students trouble are often those that took a long time to develop historically. Good mathematics teaching should not banish ambiguity, but enable students to master it. Conflating mathematics with its logical structure results in teaching that values the presentation of logical structure over understanding.

The Mathematician's Brain: A Personal Tour Through the Essentials of Mathematics and Some of the Great Minds Behind Them

  • David Ruelle
Princeton University Press: 2007. 176 pp. $22.95 9780691127385 | ISBN: 978-0-6911-2738-5

The Mathematician's Brain by David Ruelle tackles some of the same questions as Byers' book but has a different emphasis. Ruelle imagines an enormous infinite-dimensional space of all formal axiom systems and all possible formal consequences of them. The human brain, unlike a computer, is not well adapted for working directly with formal statements. Human mathematics is a discussion in natural language, as used in everyday communication, about a formalized text, which remains unwritten (but which could in principle be written). Mathematicians work primarily with ideas, which Ruelle defines as short statements in human mathematical language that can be used in a human mathematical proof. Human mathematics is a labyrinth of such ideas that allows mathematicians to move within the huge space of formal statements. Ruelle notes the extraordinary amount of context that underwrites mathematical activity, and reminds us that there is no reason to expect that a short human mathematical statement admits a short proof. The text is enlivened by many unusual mathematical examples, and by Ruelle's reflections on his own and other famous mathematicians' experiences.

The Mind of the Mathematician

  • Michael Fitzgerald &
  • Ioan James
Johns Hopkins University Press: 2007. 312 pp. $30 9780691129822 | ISBN: 978-0-6911-2982-2

If mathematics is what mathematicians do, are there any psychological traits or personalities that characterize mathematicians? Ruelle addresses this lightly with some illuminating insights, whereas Byers does not. The Mind of the Mathematician by eminent topologist Ioan James and psychiatrist Michael Fitzgerald focuses almost exclusively on this question. The book begins with a systematic survey of the literature on the psychology of mathematicians. It then surveys the literature on mathematical education and considers the psychodynamics of mathematical creativity. In both instances, the authors focus on the exceptionally able and pay attention to issues of gender. The second half of the book consists of exquisite biographical sketches of 20 prominent mathematicians. The authors' careful treatments are an especially welcome addition to a genre riddled with apocryphal anecdotes and shoddy scholarship.

All three books are useful additions to the literature and have little overlap. Those with a serious interest in the psychology of creativity should read the book by James and Fitzgerald. Mathematicians and theoretical physicists will enjoy Ruelle. Everyone should read Byers. You may not agree with him, but his lively and important book establishes a framework and vocabulary to discuss doing, learning and teaching mathematics, and why it matters.