In 1943, a mathematician named Abraham Wald was given a problem to solve by the U.S. Air Force. Its engineers wanted to know where to reinforce the armour on fighter planes to best protect them in combat, but keep them light enough to fly. And for his consideration, they showed him the planes that were making it back to home bases, their fuselages riddled with bullet holes, but with the engine area unscathed. Didn't that mean the fuselage should be reinforced?
An excerpt of Wald's subsequent calculations is included in the opening chapter of Jordan Ellenberg's new book, How Not to Be Wrong: The Power of Mathematical Thinking. But the puzzle was really solved by logic. Wald told the air force to add armour "where the bullet holes aren't." His reasoning rested on what they weren't seeing: The planes that never made it home were the ones being hit in the all-important engine area.
The central theme of Ellenberg's book is that mathematics is a worthy and essential pursuit that's necessary for revealing life's hidden puzzle pieces. He uses humour and playful sketches and a collection of fascinating anecdotes to introduce math-challenged readers to concepts such as Galton's ellipses and Berkson's fallacy. Don't be deterred if that sounds ponderous. He also explains the real value of a lottery ticket, how not to be fooled by a stockbroker's pitch, and whether handsome men really are meaner than their less-attractive counterparts (they're not).
In doing so, Ellenberg, a writer and mathematician at the University of Wisconsin-Madison, adds his voice to the growing argument that teaching mathematics primarily as a set of directions to be obeyed rather than a thoughtful exercise in critical conversation has all but banished the subject to the Ivory Tower. In a world brimming with information, math is an important tool to help spot statistical glitches and everyday fallacies, but it's being lost. "Math is the science of not being wrong about things," he writes. "Knowing math is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world."
And everyone, he argues, should have the chance to wear those specs. Ellenberg was a child prodigy who scored a perfect 800 on the math SAT at age 12, so his math teachers basically left him alone to solve his own problems. He was also one of the students followed in the longitudinal study of "mathematically precocious youth" co-authored by David Lubinski, a psychologist at Vanderbilt University. Lubinski recently lamented that these special students – "the ones who are going to figure out all the riddles" – were not getting enough special attention in school.
That doesn't jive with Ellenberg's democratic notions of mathematics – in fact, he argues against the "cult of genius" in his book. Basic arithmetic, he says, shows that Lubinski's claims about the kids who'll figure out all the riddles can't be literally correct: As Ellenberg observed in a recent Wall Street Journal essay, "most child prodigies are highly successful – but most highly successful people weren't child prodigies." Advances in math happen communally, with discoveries piling on top of each other, so the field suffers when bright kids drop out because math is seen as belonging to the "naturally gifted" and not the product of hard work – a discouraging trend Ellenberg says he sees often among his own students.
But more than that, he argues, math needs to rejuvenated as a hobby, a subject released from blackboards and textbooks and classrooms. Otherwise, he says, "it would be like nobody played the guitar around a campfire just because they weren't a professional musician."
Mathematical amateurs have all kinds of reasons to use math. It helps them learn the difference between correlation and causation, to see the flaw in statistics, to spot a sneaky sell.
"Math is the science of not being wrong." Ellenberg writes. In the real world, it doesn't just find the right answers – it teaches us to ask the right question in the first place.
How not to be fooled by a stockbroker's pitch, according to Ellenberg
Math helps to reveal what's happening behind the curtain, the story-changing details you may miss at first glance. For instance, in his book How Not to be Wrong, mathematician Jordan Ellenberg explains this parable about a stockbroker searching for new clients.
One day, you receive a letter from him, accurately predicting the pending rise or fall in the value of a stock. The next week, another letter arrives with another correct prediction. Ten weeks, and 10 correct predictions later, you'd be crazy not to hire him.
But hold on – what are you missing? Let's say the stockbroker actually sent out stacks of letters – 10,240 in Ellenberg's example – half predicting the stock would rise, half predicting it would fall. On the second week, he sends 5,119 letters, this time only to the people who received the correct prediction the first week. Half of them will think he got two for two. And on it goes, for 10 more weeks, until he's down to 10 people – including, possibly, you – who think he is a Wall Street magician.
As Ellenberg writes, he couldn't find evidence that his stockbroker scam ever actually happened. But a version is "alive and well in the financial industry," he argues, when companies test-drive new mutual funds before making them public. They quietly dump the ones that do poorly, and then pitch the ones with fabulous returns.
The stockbroker con works, Ellenberg writes, "because it doesn't try to tell you something false – rather it tells you something true from which you're likely to draw incorrect conclusions."
The more accurate conclusion from this real-world math lesson? Your odds are safer in longer-term, less flashy funds. And beware unsolicited letters from stockbrokers.