Numerical concoctions

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Once Upon a Number: The Hidden Mathematical Logic of Stories

Penguin: 1999. 214 pp. £12.99, $23 (hbk)

Sadly, we do not make our way through life the way professors of statistics, or of any other science, would have us do. Outside the classroom, most of us, most of the time, make elementary errors when reasoning about probabilities. Juries fail to give equal weight to all the evidence and cling obstinately to just those facts that suit their purposes; lawyers have been known to mislead juries over the interpretation of DNA evidence; psychics do well, though doing no better than chance; fraudsters separate the credulous from their money.

To his credit, John Allen Paulos, a mathematician at Temple University in Philadelphia, would not address these problems by herding us all back to school or college. He thinks the problem lies in the way we build up the stories we use to get us through life, the explanations we concoct and the way we feed quantifiable information into the mix. In this book he wanders around the issue, hoping to shed what even he calls an oblique light on the matter.

Sometimes the issue is straightforward; seemingly very rare occurrences, in fact, crop up all the time. We have a bias towards blaming people when someone is injured, but saying it was only bad luck when an exactly similar mistake injures no one. In such cases, our prior views about the way the world operates twist our use of the numbers.

Everyone uses rules of thumb to deal with the complexities of real life. These may have something going for them, even if they wouldn't get good marks on a statistics course, because they allow us to filter out the complexity. Here, however, the author may not appreciate the scale of the problem. At one point Paulos jokes (and the book is full of jokes, some of them very funny) that the typical statistics problem vexes thousands and pleases seven or eight. In fact, most of his examples are contrived in this possibly vexatious way. This is partly because Paulos is writing a popular book in which the arithmetic must be made easy and the assumptions kept simple. But it is partly a function of the way statisticians operate, because the complexities of real life are daunting. The end result, nonetheless, is hypothetical situations that are too simple to convince us that their statistical morals apply to real life, and it is not always clear that Paulos realizes this.

Statistics is not the only discipline to take a toy fishing-net to the ocean. The second half of the book is at least as interested in the disparity between simplified pure logic and the ways we reason, which various novel accounts of logic attempt to describe. Paulos mentions situational logic in this context. He also considers the complexity of the world, describing how complexity theorists think about it; he alludes to Ramsey theory, and generally discusses how order and simplicity always seem to turn up somewhere.

These are important issues, and the reader who does not know of this work will be amused and diverted. But the claim that this connects to our use of stories is a tenuous one. It is not that anything Paulos says is wrong, so much as that everything he says is superficial. There are deep problems here about order, chaos and complexity, on the one hand, and how our mind works, on the other, but on the evidence presented here we are very far from putting the two together.

Paulos intends the term ‘story’ to mean more than just the little analyses we make up for ourselves. He wants it to reach as far as published stories, from romantic and detective fiction to the profundities of Anton Chekhov. There is nothing here that will help the reader enjoy fiction more, read it more carefully or understand how it works. We are indeed dangerously close to the literal-minded approach to fiction so ruthlessly parodied under the heading, “How many children had Lady Macbeth?”.

The book comes nowhere close to living up to its subtitle. It does not provide a mathematical logic for stories, but merely suggests that intentional logic addresses the issue of finding meanings in the confusing world around us. It does convey a good sense of the difficult nature of that task, at which humans manage to be remarkably good while being in some basic ways rather bad. There is probably somewhere an automatic theorem-proving machine that can outstrip most of us at solving puzzles of a logical or probabilistic kind. But if so, that would only prove yet again that computers can do some things very much better than we can, and still can't think. Resolving this conundrum will help us understand how the mind works. Until then, books like this one will tell us, very enjoyably, that there is work to be done.

More on numbers

What is Random? Chance and Order in Mathematics and Life by Edward Beltrami Springer, $22, £15.50

Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems, and Musings edited by William Frucht Wiley, $27.95, £22.50

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Gray, J. Numerical concoctions. Nature 402, 724–725 (1999) doi:10.1038/45378

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