Wednesday book: Maths: a matter of life and death

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The Independent Culture
Once Upon A Number: The Hidden Mathematical Logic Of Stories

By John Allen Paulos, Allen Lane/Penguin Press, pounds 12.99

JOHN ALLEN Paulos is the maths teacher I found 25 years too late. His outstanding first book, Innumeracy, was not only great fun but argued compellingly that many of the problems we face in everyday life - from belief in astrology to assessment of risk - were products of our ignorance of basic mathematics. We all accept how essential it is to know how to read. Paulos compellingly argued that a grasp of basic aspects of, for example, statistics and probability, was essential in order to "read" experience.

Despite the misleading title, Once Upon a Number explores much the same territory. What is the connection between our own subjective experience of the world and the abstract, statistical analysis of it? Paulos's argument here, as in Innumeracy, is that statistical analysis - however technical - is not as abstract as it seems. The context of any set of statistics is crucial - not least as a way of assessing its relevance and accuracy.

This subject may sound abstract but its practical effects can be tragic. The British Medical Association Guide to Risk concludes its assessment of smoking by saying that smokers consider it "an acceptable risk, but in practice the risk they are prepared to accept is assessed by them as being a lot lower than it really is". Or think of the current government banning beef on the bone in response to a projected possible death rate of one person a year. Jack Cunningham defended it by saying: "Yes, but what if you were that one."

Much of the pleasure here lies in the specifics, even when they make uncomfortable reading. Paulos cites the study by a group opposed to capital punishment that showed that blacks convicted of murder in Philadelphia faced four times the odds of being sentenced to death as others convicted of the same offence. This sounds appalling but, as Paulos (himself a liberal) shows, using the odds of an event (the probability of an event occurring divided by the probability of it not occurring) greatly overstates it. If the figures for death sentences were 99 per cent for blacks and 96 per cent for others, that would come out at four times the odds. Unjust, of course, but not as drastically as the original description suggests.

Some of the most enjoyable parts are only distantly related to the main theme. A section devoted to the role of humour in computation features what he claims to be the true story of a well known philosopher delivering a talk on linguistics. He had just stated that the double negative construction in some natural languages has a positive meaning and in other languages a very negative meaning. He went on to observe, however, that in no natural language does a double positive construction have a negative meaning. At this, another philosopher jeered "Yeah, yeah".

There are some enjoyable brain-twisters that demonstrate the role of logic in normal life. There are are four cards lying on a table each with a letter on one side and a number on the other. Face up are D, F, 3 and 2. Which should you turn over to prove the statement that if a card has a D on one side, it has 3 on the other? Most people choose D and 3 but ought to choose D and 2.

As always, Paulos is instructive about statistical analysis and the dangers of applying it ignorantly. In this book he takes it further and looks at the paradoxes of applying logic in the complicated world of human interaction. He cites a troubling example from the game of poker, in which hands are ranked in order of the probability of getting them. Three of a kind is less probable, and hence more valuable, than two pairs. However, if you play jokers as wild cards, then three of a kind becomes more likely than two pairs. But if you adjust the rules to make two pairs more valuable than three of a kind, then two pairs will become more common. As Paulos says, this example reveals a lot about why everyday life can be so complicated (Penguin has already published a better book on this subject: Labyrinths of Reason, by William Poundstone.)

For much of the time in this rather loosely constructed book, Paulos uses "stories" to mean our way of interpreting the world, which is fine. Occasionally he stumbles into the subject of stories in literature, but the relationship of literature to the world, and the difference between good and bad literature, is so complicated that he really has nothing to contribute.

This is another captivating read. Sometimes it can seem as if Paulos is deploying the same argument in book after book. But then you see another astrology column or hear Blair and Hague exchanging statistics, and you think it can't be written often enough.