Richard M. Gillette.
Mathematical Sciences
MSU-Bozeman
Does a mathematician's reading of the newspaper qualify as a topic for a book? In this instance, perhaps so. Tohn Alien Paulos, a mathematics professor at Temple University, is best known as the author of "Innumeracy: mathematical illiteracy and its consequences", a book which promoted a national educational concern and gave life to an aptly coined addition to the language. In this book, Paulos skillfully lectures the reader on the mathematical and statistical features of the stuff of daily life as reflected in newspaper headlines. The author's mission is to convey a sense of what is lost to the innumerate consumer of the daily media, and, occasionally, to afford a glimpse of contemporary mathematics.
The five sections of this book are organized by contrived-but-plausible newspaper headlines (the book contains 54 short articles if the section introductions are counted). Follow the author's design: read this book as you would a newspaper. Scan the headlines, go with your interests, and perhaps you will be rewarded. The mathematics you will encounter is mostly the common sense mathematics that belongs to us all -- a mental discipline learned as a folk art by solving puzzles and playing cards and board games. In the words of the author: "Mathematics is not Primarily a matter of plugging numbers into formulas and performing rote computations. It is a way of thinking and questioning that may be unfamiliar to many of us, but is available to almost all of us."
In the piece headlined "Pakistanls Bhutto Gambles in Trade Negotiations: On Dice and Bluffing", a couple of nicely presented optimal strategy problems illustrate the "value of acknowledging uncertainty and behaving accordingly". In a spinner game, the player is to predict in advance of each spin whether the spinner will land on red or black. If the spinner is biased, landing on red 70% of the time and landing on black 30% of the time, how should the player guess on each spin in order to maximize the number of correct guesses over the long haul? If you are thinking that the player should guess red more often than black you are wrong, as the author carefully explains. For example, guessing red 70% of the time leads to only 58% correct guesses in the long run because on average only 70% of the spins which turn up red are predicted correctly, and only 30% of the spins which turn up black are predicted correctly (70% of 70% plus 30% of 30% comes to 58%). But guessing red 100% of the time surely leads to 70% correct guesses in the long run -- no mathematical formulas required for that deduction.
It is a bit of a stretch from biased spinner game to Bhutto's gamble, but the point of the piece is this: It is necessary to come to terms with randomness in devising optimal strategies, and an optimal strategy may in fact be based.on a 'gamble'. Paulos gives a more complicated example of an optimal strategy for an idealized baseball game between a batter and a pitcher in which the pitcher has two pitches -- a fast ball and a curve ball. The batter's averages depend on the pitch thrown and on the batter's expectation for the pitch. Against fast pitches the batter averages .500 when expecting a fast pitch, but only .200 when expecting a curve. And against curves, the batter averages .400 when expecting a curve, but only .100 when expecting a fast ball. The surprise is that in these circumstances there is a strategy which is simultaneously optimal for both pitcher and batter. The optimal strategy would have the pitcher randomly pitch fast balls half the time (the pitcher could toss a coin before each pitch), while the batter should plan on fast balls one-third of the time (the batter could roll a single die before each pitch). The batter's average under the optimal strategy is expected to be .300. There is a bit more than common sense mathematics in this example, which uses methods from the theory of two- person games, but the author avoids technical discussion in favor of arguments grounded in intuition.
The segment "Ranking Health Risks: Experts and Laymen Differ", which originally appeared in the March, 1994 issue of Discover magazine, is one of the best in this book. This essay raises questions about the way in which our society assesses risk. Society is most distressed by the scourges of cocaine and heroin use which cause 14,000 deaths per year, while alcohol and tobacco use cause 490,000 deaths annually. We worry about the risks of pesticide ingestion unaware of the fact that we ingest much more naturally occurring pesticide than man-made pesticide. Perhaps the most important issues raised in this essay have to do with the uses and misuses of probability and statistics (Paulos is a probabilist by training). Paulos uses the artificial but realistic example of a medical test for a rare (one in a thousand) disease. The test is supposed to be 99% accurate, yet, as the example makes clear, a person who tests positive for the disease actually has only about a 9% chance of actually having the disease.
This same article illustrates one of this reviewer's Problems with this book: there are topics here which demand more than they are given. For example, it is one thing to numerically blow away a misconception such as the relative risks of tobacco versus cocaine. It is quite another thing to leave the impression that this simple numerical scoring assesses the "real hazard". It is possible (who knows?) that the number of deaths from cocaine use might be much higher had the use of cocaine not been proscribed on account of a perceived risk. To take another example, the Delaney Clause, which was used by the FDA to Prohibit food additives which cause cancer in test animals, has been opposed by industry on the grounds that often the test amounts which induce cancer in animals are very large, so that the real risk to humans might be very small. The Delaney Clause allowed no leeway for carcinogenic additives. Paulos leaves the impression that because it is practically impossible to reduce a small risk to zero, the passage of the Delaney Clause was based on unfounded or excessive risk assessments. It would have been much more interesting to read a discussion of the difficulties in extrapolating from high-dose risk down to a threshold risk. (Congress has done away with the Delaney Clause, so the low-dose testing will now take place in the human population.)
To mention another example, the headline "Lani 'Quota Queen' Guinier" alerts the reader's political antennae to a level of discussion which is not attempted in the accompanying article. In the words of the author "... rather than rehash the ideological aftermath of the political fray, let me describe a simple mathematical idea that motivates some of Professor Guinier's writings". What follows in the article is a quite interesting discussion of something called the Banzhaf power index, as well as a nice account of cumulative and approval voting schemes. Unfortunately, the politically literate reader may be hung up on the first part of the sentence quoted above: "I'm sympathetic to (most of) the aims of the Voting Rights Act, yet strongly opposed to quotas (whether they're called that or not) - ...". It would have been interesting to read the author's definition of "quotas" and to discover the connection between quotas and cumulative voting. (On this subject, a recent Washington Post article showed the outlines of two tortuously convoluted Texas congressional districts and invited the reader to guess which is illegal. Though both districts appear bizarre in shape, the majority white district is legal; the majority black district is illegal.)
Paulos' articles are not uniformly successful. A successful article is "More Dismal Math Scores for U.S. Students" which is extracted from an article in the Washington Post. This essay isolates some of the most commonly held misconceptions about mathematics clearly and concisely. Quoting the author: "Probably the most harmful misconception is that mathematics is essentially a matter of computation. Believing this is roughly equivalent to believing that writing essays is the same as typing them".
By contrast, the segment entitled "Asbestos Removal Closes NYC Schools" tortuously makes a simple point which had already been made more than once in earlier articles, and "GM Trucks Explode on Side Collision" entitles a segment which seems to add little to the fashionable but tiresome criticisms of product liability law suits and their symbiotic media coverage.
On balance however, I think we should be pleased that Prof. Paulos has written this book. It touches on many areas--game theory, graph theory, non-linearity and chaos in population models, risk assessment, Smale's horseshoe, Arrow's theorems (regarding voting methods), the central limit theorem, conditional probabilities, complexity theory, mathematical logic, and more. There is surely something here for everyone.