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July 29, 2014 8:36:02 AM

Now it is time to read a book about mathematics. That sentence in itself may have put you off. Mathematics for many of us is something we endured in school, and since we can calculate a tip or balance a checkbook, we just don't need much more. Mathematicians, we think, are engaged in their own world, building proofs upon proofs that do not have much to do with the way the rest of the world works. Jordan Ellenberg is mathematician. His work is in research into number theory, the beautiful and arcane behavior of those mysterious entities, the whole numbers 1, 2, 3, and so on. Number theory has proved surprisingly practical in encryption, but that's not the sort of message Ellenberg wants to get across in *How Not to Be Wrong: The Power of Mathematical Thinking* (The Penguin Press). Mathematics is not an esoteric language that happens to be useful when experts apply it. Ellenberg explains that he and his mathematician buddies do spend most of their time in the profound and complicated areas of mathematics. "That's where the celebrity theorems and conjectures live: The Riemann Hypothesis, Fermat's Last Theorem, the Poincare Conjecture, P vs. NP, Godel's theorem..." There are books that try to explain each of these to the non-mathematician, but Ellenberg is more interested in showing that profundities do not have to be complicated and that looking at things in a mathematical way makes them more meaningful. "Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world... With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way." The chapters here all show the beauty of attempts to understand how mathematics bears on the everyday, and Ellenberg is a great explainer, with good examples and good humor. It isn't always easy; there are pages where you cannot just let the words sink in but have to think hard about some of the computations. The enjoyment of sharing Ellenberg's appreciation for his subject, though, is well worth the effort.

Mathematics is good at making models and good at making sure those models are not simplistic. Take, for instance, the question of welfare programs in the US. Many commentators are pointing to the example of Sweden which is an archetype of the social welfare state but is trimming such programs down. Both the US and Sweden are trying to maximize prosperity; surely if Sweden is progressing toward fewer social services in furtherance of that goal, it would be foolish for the US to be trying to be more Swedish by increasing such services. The line between prosperity and tax-funded social services is a straight one, in such a view, connecting one extreme point of low social services and high prosperity to another extreme point of high social services and low prosperity. But what if this is the wrong model? What if the line is an inverted U-shape, with prosperity increasing to a maximum as social services increase, and decreasing thereafter? Sweden and the US might be seen as being on opposite sides of this curve, and thus taking respectively opposite actions to get nearer to the maximum. There are plenty of relationships like this that are not linear, and the direction one should go should be based on where one already is.

Put that way, it seems like common sense, and one of the themes of this book is that mathematics takes common sense and gives it a power beyond the obvious. Another example here is of Abraham Wald, a mathematician faced with the problem of designing more robust airplanes during WWII. You can coat an airplane in armor, but it costs money to do so, and makes the airplane heavier and less useful. So the idea is to put the armor only where it is going to do the most good. Officers documented that planes returning from missions had more bullet holes in such places as the wings and the fuselage, and not so much on the engine. With more bullets hitting the wings and fuselage, they thought that those were the areas that needed the extra armor. Wald was consulted, and thought about the problem in a different way. The way Ellenberg tells the story, it sounds like common sense, but he includes a page of Wald's actual calculations to make his re-thinking of the problem rigorous (amusingly, in the included page, Wald has admitted, "The exact calculation is tedious..."). What Wald demonstrated is that bullets would be hitting the airplane all over, randomly distributed; there was no reason to think that wings were particularly liable to take a hit other than that they had a big area to be hit. And Wald's insight was that the returning planes had more bullet holes in the fuselage and wings because they were able to return with hits to those areas. The sample of planes returning did not include those planes that had been shot down, and those were the ones that weren't taking the superficial wing or fuselage hits, but hits to the engines. Any armor that needed to be applied, then, ought to protect the engines.

Though this seems like mathematics backing up common sense, sometimes common sense didn't come into play until mathematics made it common. In the discussion of expected value, Ellenberg cites Britain's scheme of life annuities to gain capital in its 17th century wars. People who purchased this sort of insurance paid a set fee; grandparents and infants paid the same amount. Even to those of us who are not actuaries, this does not make any sense; for basic life insurance, a lump sum for an infant ought to be a lot more than for a grandparent, since the years of coverage will be so much more. This was not common sense at the time, however. It took Edmond Halley, the Astronomer Royal, to demonstrate, using birth and death statistics, the probable life spans for anyone proposed for insurance, and now it all seems obvious. (Yes, this was the Edmond Halley of the comet, who shows up another time in this book as the man who drew maps with connected curves showing local discrepancy between true north and what the compass said was north. Halley made contributions in astronomy, meteorology, physics, and more. As Ellenberg says, "This guy really knew how to keep busy between comets.")

Ellenberg also shows that there was a time when no one thought mathematics had anything to say about probability. Probability theory, which is often counterintuitive, shows up here plenty, especially in the chapter on lotteries. How do you keep from losing money on Powerball? Just don't play. The calculations here show how buying a lottery ticket is simply a bad risk, although if you get a kick out of a bad risk now and then, Ellenberg shows how you ought to play only when the jackpot is really big, and you ought to decrease the risk of having to share the pot with someone else by picking numbers that are not birthdays and are not from a fortune cookie. But then he goes on to show how there have been state lotteries run in a certain way, with a technical glitch that enabled syndicates to buy up huge numbers of tickets and have nearly guaranteed profits thereby. (A mathematical pal of Voltaire figured a similar way of gaming the lottery, enlisted friends, including Voltaire, to buy up stacks of tickets, and made all of them rich. "What - you thought Voltaire made a living writing perfectly realized essays and sketches? Then, as now, that's no way to get rich.") Ellenberg ties the abstract branch of mathematics, projective geometry, to the story. It has no obvious connection to probability, but it can be used to show what numbers the syndicate ought to concentrate on. It's not the only time that there seems to be a mystical connection between unrelated mathematical realms.

Ellenberg's book is divided into five different, overlapping sections: Linearity, Inference, Expectation, Regression, and Existence. To illustrate the array of subjects, and the humorous tone, here is what is included in the Regression section: "Hereditary Genius, the curse of the Home Run Derby, arranging elephants in rows and columns, Bertillonage, the invention of the scatterplot, Galton's ellipse, rich states vote for Democrats but rich people vote for Republicans, 'Is it possible, then, that lung cancer is one of the causes of smoking cigarettes?,' why handsome men are such jerks." It's a useful tour of a lot of a lot of material we non-mathematicians do not think about very often, and a reminder of the sway mathematical patterns have over broad aspects of our existence.