The Power and Fun of Mathematical Thinking

How Not To Be WrongHow Not To Be Wrong by Jordan Ellenberg is a fun book to read; perhaps surprising since the topic is mathematics. Ellenberg begins by saying the seemingly pointless drills primary school students complain of are akin to practice in sports. This hooked me immediately, since I think too many people believe you somehow “understand” math when you read a textbook and then can “do it”.  Ellenberg says “if you want to play soccer… you’re going to be spending lots of boring weekends on the practice field. There’s no other way.” It’s the same with math. An ability to perform basic operations is important to thinking since, as he observes, it would be hard to write a sonnet if you had to look up the spelling of each word as you worked.

Ellenberg does object to some of the way math is taught. Calculating “is something a computer can do quite effectively. Understanding whether the result makes sense – or deciding whether the method is the right one to use in the first place – requires a guiding human hand… A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.”

His engaging style is evident throughout the book. I laughed out loud several times, and I urge you to read the footnotes – they’re often funny. For example, when introducing Leonard Jimmy Savage, a pioneer of decision theory and Bayesian statistics, Ellenberg adds this footnote: “Savage… at one point spent six months living only on pemmican in order to prove a point about Arctic exploration. Just thought that was worth mentioning.”

Savage worked on a World War II problem for the US.  The Navy thoroughly surveyed how many bullet holes were located in each part of each plane returning from battle, and wanted statisticians to confirm more armor(too heavy to cover the whole plane) should go on the areas with the most holes. But they were convinced to armor the areas with the fewest holes. Why? Because planes hit there failed to make it back to be surveyed. The Navy’s error was assuming the planes they surveyed were representative.

Ellenberg covers a lot of statistics and probability. I especially liked the section on straight line extrapolations, which are so easy to do and so often wrong. He comments on one report that predicts 100% of Americans will be obese before 100% of black American men will be obese; and no one thought the two results looked odd together.

Ellenberg offers sections on prime numbers, the Laffer curve, Bible codes, the hot hand in basketball, and haruspicy (reading the future in sheep entrails). He demonstrates how ignorance of the Law of Large Numbers leads the government to both celebrate and condemn small schools when the wide variability in their students’ performance (compared to large schools) is mathematically inevitable. He even tackles how math and God relate. It is a long book that ambles through many topics, so plan some time to read it.

I should point out that negative reviews on Amazon generally complain that Ellenberg treads too far into politics. To combine a couple reviews: “hoping to see a sterile mathematical analysis” but instead “a very strong socio-political agenda, which seems inspired by the author’s domicile in Madison, WI – the hotbed of mainstream academic liberalism.” Ellenberg’s analysis of the Bush vs Gore election and how flipping a coin might be a better way to end tied elections comes in for criticism. Some reviewers wanted a straight telling of mathematics without getting sidetracked into stories, and some recommended Freakonomics instead.

Uncertainty is part of life, and contrary to many people’s notions, mathematicians do not always seek a single, definitive answer to an abstract question. “Uncertainties percolate through… [and] feedback into each other.” At some point, the noise engulfs the signal and insisting on “an answer” is wrong. But “the quibblers and the naysayers and the maybe sayers don’t make things happen”; “uncertainty and revelation can mingle… Not being sure is the move of a strong person…and math is part of it… Math gives us a way of being unsure in a principled way… making a firm assertion: ‘I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.’ Or even more: ‘I’m unsure, and you should be too.'”