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A Lens of Numbers

Malcolm Sherman

The Magical Maze: Seeing the World through Mathematical Eyes. Ian Stewart. John Wiley, 1998. $24.95.

Ian Stewart is a master expositor with an encyclopedic range who gives his readers much food for thought. These elegant essays grew out of the author's 1996 Royal Institution Christmas Lectures for young people, the 168th in a series begun by Michael Faraday in 1826. Stewart's previous popularizations led to his receiving the Michael Faraday Medal of the Royal Society for contributions to public understanding of science.

Stewart begins with the simplest puzzles (party tricks that begin "think of a number"), progressing rapidly to deeper issues with applications and connections that will surprise and delight readers. For example, in a very large Pascal's triangle (whose entries are binomial coefficients familiar to high school students) almost all the numbers are even and the pattern of odd numbers is well approximated by the 20th-century fractal object Stewart describes, called the Sierpinski gasket.

Although chaos and fractals and other broad topics have appeared in Stewart's previous books, his accounts do not overlap. The maze metaphor means that both a mathematical argument and an author's choices are logically constrained paths through mazes of possibilities.

The "passage" (Stewart's word for chapter) "Queen Dido's Hide" is a far-ranging treatment of optimization problems. Stewart gives a lucid account of Jacob Steiner's 1838 proof that among all curves of fixed perimeter, a circle has the largest area. He also explains that the original proof was incomplete because Steiner assumed that a solution existed. He then asks which lacing pattern minimizes shoelace length, noting that Europeans typically tie shoes differently than Americans. This example deftly illustrates his broad theme that mathematicians see mathematics in everyday objects.

Stewart expects some readers to question his accounts of the "Interrogator's Fallacy" and the "Prosecutor's Fallacy." The former is the assumption that a confession by an accused criminal always increases the probability of guilt. On the contrary, says Stewart, hardened criminals, including terrorists, are less likely to confess than the innocent because they are trained to resist police coercion. Stewart's abhorrence of torture would of course be justified even if forced confessions are actually associated with an increased probability of guilt, perhaps because the police are more likely to select the guilty for abuse. The "Prosecutor's Fallacy" is the confusion between two probabilities—for example, the probability that an innocent person's DNA would match that found at the scene of a crime and the probability that someone whose DNA matches the crime scene sample is innocent. But statisticians, who do understand this distinction, are still willing to reject a null hypothesis when the P value is small enough, even though they don't know the probability that the alternative hypothesis is true.

The passage on probability, "Marilyn and the Goats," begins with the well-known problem posed by columnist Marilyn vos Savant: After a game-show contestant has made an initial choice among three gates, only one of which leads to a prize, the host always provides the additional information that one of the remaining gates leads to a goat—not to the prize. Does the contestant increase his chances of winning the prize if, after one gate is thus eliminated, he changes his initial choice? Stewart's explanation of why the contestant should change is quite clear, but may still not convince the thousands who wrote to Marilyn to disagree with her published solution. Stewart reports that Marilyn had lots of fun deriding her academic critics, who unfortunately included Ph.D's in mathematics.—Malcolm J. Sherman, Mathematics and Statistics, The University at Albany-SUNY

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