John Allen Paulos is a worldly mathematician, subspecies logician, who has noted with dismay the extreme level of naivete concerning quantitative issues in public discourse and thought. A decade ago he wrote of "Innumeracy" in a book of that title; the coinage a parallelism to "illiteracy." Via a collection of examples and mathematical vignettes, the book exposed a general public ignorance in quantitative and logical knowledge. Perhaps to his surprise, assuredly to his pleasure, his book reached an appreciate audience, and he has continued to express himself through books, columns and lectures. The present book was published late last year. Many Paulos devotees have already found the book and need no review. This review is for others.
This book is Paulos' "response to the relative merits of narratives and numbers," that is, between the specific and personal and of the general and detached, "the two fundamental ways of relating to our world." Both terms are broadly meant. "Numbers" includes statistical science, other quantitative reasoning, formal logic, and some science in general. "Narratives" range over descriptions of events--from news stories to case studies to gossip--to informal reasoning, tales and literature. Thus his vision is wide, if idiosyncratic. "What are the narrative implications of the mathematical notions of complexity and `order for free?' What do interpretations of literature have in common with applications of statistics? What does literary criticism have to do with cryptology?" He ranges further afield than with his earlier books, into fields in which he is an amateur, and at times it shows. At times he admits it shows. Read with this in mind, the book is rewarding. As a penetrating study, the book is less successful.
It is difficult to summarize the contents of the book. Paulos is not one to spend the time to deeply analyze a point. "Indeed, this book's piecemeal, episodic structure is due to a similar impatience with lofty claims and simplistic theories." Seldom does a particular topic remain for more than two or three pages, whence the book zigs or zags to something new. To open a few pages at random: the Simpson trial, Ramsey theory, "many worlds theory" in literary analysis, Turing tests, the Asian currency collapse, complexity horizons, and Presidential affairs. The book was written before the political events of the past season, but Paulos points out that sometimes it does matter what your definition of "is" is. In fact, Paulos would enjoy the difficulty in summarizing; in complexity theory, the measure of information in a message is the inability to shorten it.
The title of each chapter save the last sets up a dichotomy between narrative and number. Thus "Between Informal Discourse and Logic," "Between Meaning and Information," etc. Here, logic, part of number, means formal extensional logic, and information means quantitative theories of information. The broad theme is that these are complementary; although sometimes narratives and number clash, they are both essential to our understanding of reality, and a person cannot be ignorant of either. His book is a collection of illustrative examples, viewed from his bent and bias. He is at his surest early on, where he is closest to his expertise and to the subject matter of earlier books. He begins by contrasting "stories and statistics." To illustrate with an example Paulos does not use: the issue of illnesses occurring in clusters in a localized region, presumably caused by earlier chemical pollution of the ground. As narratives, the cases are compelling. Abstracted to statistics, the situation becomes extremely murky, and it is not at all clear what reality will emerge. This issue is current and consequential, and a careful exposition of the issues would be a public service. However, the mathematics could get technical and inconclusive and Paulos' exposition works best when his mathematics is crisp-wrapped up in a bow.
Readers of Paulos know he is at his best conclusively elucidating the issues around a point of statistics or logical fallacy. A case from the book of Paulos at his best: bringing Bayes' Theorem, which tells how to compute probabilities in the light of more-complete information, to bear on O. J. Simpson's defense attorney Alan Dershowitz' repeated claim that since fewer than 1 in 1,000 women who are abused by their mates go on to be killed by them, the spousal abuse in this case was irrelevant. True data, but a non sequitor, claims Paulos. The more-complete information is that Nicole Simpson was in fact killed. With Bayes' Theorem, the computation is that if a man abuses his wife or girlfriend and she is later murdered, the man is the murderer more than 80% of the time. "Statisticide," Paulos calls it.
His ground is mushier later in the book when narrative comes to mean literature. Here he intends to bring mathematical reasoning to bear on the meanings of fictional works. He touches on some modern theories of literary criticism, and the reader will get a nice exposition on extensional versus intensional logic--Paulos is a logician--but to this reviewer, this part of the book reads best if it is understood that these are musing of an outsider. He brings new viewpoints, but his musing are untested.
However, a reader attuned to Paulos will find nuggets throughout the book. Particularly on points of reasoning, he has a way of isolating the essential detail. He can stretch the consequences rather thin, however. He makes an incisive point about how common knowledge is not useful until it remarked that it is common--in his example all the wives of a village kill their philandering mates--however, it seems a bit of stretch to attribute the collapse of Asian currencies last year to same phenomenon. Is God to be defined in terms of complexity theory? Paulos reprints a speech he gave on mathematicians' humor--not an oxymoron. He is not immune himself, and the reader should watch for the whimsy, the self-referent jokes, the meta-gags, the numerate puns.
A fair amount of the mathematical material is recycled from Paulos' earlier books--it would be unfortunate if this well has run dry--his interest here is to move it to unbeaten paths. He writes, "I am aware that part of what is written here may be dismissed as an unholy mixture of discordant fields; even I think this on Tuesdays and Saturdays. Nevertheless, on the other five days I think it's well worth a scientist's effort to try to explore the borderland between these disparate cultures." Reader, choose your day.