Here's a cautionary tale. A broker sends you 10 free stockmarket predictions in a row that all come true, and then asks for money for an 11th. The stockbroker's offer seems reasonable enough. His strike rate is 10 out of 10, so surely he will be right the next time, too? Don't do it! If you pay for the 11th tip, you will have fallen victim to the oldest scam in the book. Unknown to you, the stockbroker has been sending every combination of possible predictions to tens of thousands of other people, and you are one of the unlucky few who got 10 good ones in a row.
Jordan Ellenberg calls this story the parable of the Baltimore stockbroker. In the UK it was popularised by Derren Brown's 2008 TV programme The System. Brown sent a woman five correct tips for five horse races, and then persuaded her to gamble her life savings on race number six. He revealed at the end of the show that he had fooled her: he had also sent out tips to thousands of others with the other possible combinations of winning horses. His failsafe system of prediction was nothing but a con.
Ellenberg writes that by explaining the ruse, Brown was "probably doing more for maths education in the UK than a dozen sober BBC specials". The crucial message is that we must always consider the bigger picture, especially when dealing in probabilities. It's a valuable lesson, since, even though there probably never was such a stockbroker from Baltimore, in spirit he is alive and well in less blatantly fraudulent but equally misleading ways.
Imagine you are considering investing in a mutual fund. Before funds are opened up to the public, they are often monitored in-house. Funds that don't perform well can be shut down, without the public ever knowing. Ellenberg warns that if you are seduced by funds with an eye-popping rate of return, then you are walking into the Baltimore stockbroker's trap: "You've been swayed by the impressive results, but you don't know how many chances the broker had to get those results."
The Baltimore stockbroker also appears in science, where the phenomenon is known as the "file drawer problem". Ellenberg shows us expertly how having a threshold for statistical significance leads to scientists "running the con on themselves". Results that fall just below the threshold are filed away in drawers, and results that are just over get published in journals. The scientific community only gets to hear about the "statistically significant" result – it never knows how many times the same experiment has failed.
How Not to Be Wrong is full of simple yet deep insights that encourage clear thinking about many areas of modern life, often with a political riff. Take this headline from a right-wing blog: "Why is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?" Translation: why should America move towards a Scandinavian model of higher taxes and bigger government when the Scandinavians are retreating from that model?
Ellenberg punctures this argument with a simple back-of-the- envelope sketch. Imagine a graph where the vertical axis measures prosperity and the horizontal axis measures Swedishness. Now draw an n in the graph, which shows that, as a country becomes more Swedish, its prosperity grows until reaching maximum prosperity; and yet, as it becomes more Swedish, its prosperity drops. If the US is a point on the left section of the curve, then becoming more Swedish is desirable, and if Sweden is a point on the right, then becoming less Swedish is also desirable. If you correlate prosperity and Swedishness with this curve, then, without contradiction, it can be desirable for one country to increase Swedishness and for another to decrease it. Ellenberg's lesson here – "which way you go depends on where you are" – may sound obvious, but too often we fail to think about the world in this way.
How Not to Be Wrong is a cheery manifesto for the utility of mathematical thinking. Ellenberg's prose is a delight – informal and robust, irreverent yet serious. Maths is "an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength," he writes. Doing maths "is to be, at once, touched by fire and bound by reason. Logic forms a narrow channel through which intuition flows with vastly augmented force."
Ellenberg is certainly pyrotechnic within his own field. A child prodigy – he scored full marks in his maths Sats aged 12, and twice won a gold medal at the International Mathematics Olympiad – he is now a professor of maths at Wisconsin university, where he specialises in number theory. How Not to Be Wrong, however, concentrates on statistics and probability, areas of mathematics for which there are more obvious applications. You will be hard pushed to find a sharper and more readable exposition of standard mathematical concepts such as p-values, expectation and correlation than in this book.
Ellenberg also writes about what life is like as a professional mathematician, and I enjoyed his observations of the academic world. He laments that maths is plagued by the genius cult, which tells students that it's only worth pursuing the subject if you are going to be the best. "Athletes don't quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was 'ahead' of them."
Most of the anecdotes that he uses to illustrate the mathematical ideas he is writing about concern the US, but this should not put off British readers, as the points he is making are universal. My one grumble is his overuse of humorous footnotes, which detract from the flow of the text.
Still, How Not to Be Wrong is an impressive work of popular mathematics. It's low on formulae and numbers, and big on ideas. From mathematics we learn that "there is structure in the world; that we can hope to understand some of it and not just gape at what our senses present to us, [and] that our intuition is stronger with a formal exoskeleton than without one." Read this book and you'll discover how.
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