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*How Not to Be Wrong* gives us an intimate glimpse into how mathematicians think and how we can benefit from their way of thinking. It also explains how easily we can be mistaken when we apply mathematical tools incorrectly, and gives advice on how we can instead find correct solutions.

- Anyone interested in math
- Anyone interested in logic or philosophy
- Anyone interested in seeing the equations behind everyday situations

Jordan Ellenberg is a professor of mathematics at the University of Wisconsin-Madison. His work covers a wide variety of mathematical topics, including arithmetic geometry and number theory. Ellenberg writes the popular column, “Do the Math” for *Slate,* and has had work appear in *The* *New York Times*, the *Washington Post* and *The* *Wall Street Journal.* He is also the author of the novel *Grasshopper King*

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*How Not to Be Wrong* gives us an intimate glimpse into how mathematicians think and how we can benefit from their way of thinking. It also explains how easily we can be mistaken when we apply mathematical tools incorrectly, and gives advice on how we can instead find correct solutions.

Convoluted mathematical formulas you encountered in school might have made your head spin. At the time, you might have asked yourself, “Will I ever use this in real life?”

The short answer is yes. Math is a key tool in solving common problems. We all use math every day, but we don't always call it “math.”

In essence, mathematics is the science of *not being wrong*.

Consider this example: During World War II, American planes returned from tours in Europe covered in bullet holes. Curiously, a plane’s fuselage always had more bullet holes than did the engine.

To better protect the planes, military advisors suggested outfitting the fuselage with better armor. One young mathematician suggested instead improving the armor for the engine.

Why? He suspected that those planes that took shots to the engine were actually *those that didn’t make it back*. If the engines were reinforced with better armor, more planes might survive.

There's a mathematical phenomenon known as *survivorship bias *underlying this situation. Survivorship bias is the logical error of concentrating on the things that “survived” some process. In this example, advisors concentrated incorrectly on the state of the planes that survived, overlooking the planes that didn't.

This example may not seem like a math problem, but it is. Math is about using reason to *not be wrong* about things.

Math is also based on common sense. Can you explain why adding seven stones to five stones is the same as adding five stones to seven stones? It’s so obvious that it's difficult to actually explain.

Math is the reflection of things we already know intuitively. In this case, math reflects our intuition by defining addition as *commutative*: for any choice of a and b, a + b = b + a.

Even though we can't solve entire equations with our intuition, mathematics is *derived* from our common sense.

- Mathematics is the science of not being wrong, and it's based on common sense.
- Linearity allows us to simplify mathematical problems.
- Drawing conclusions from observational data is questionable, but probability theory can help.
- Probability theory tells us what to expect from a bet, but we still have to consider the risks.
- The regression effect can be found everywhere, but often it isn’t recognized.
- Linear regression is useful, but assuming linearity when it isn't there can lead to false conclusions.
- Many research findings are wrong because of misused data or incorrect probability calculations.
- Polls and elections that make statements about “public opinion” are often incorrect.
- Final summary