How Not to Be Wrong Book Summary - How Not to Be Wrong Book explained in key points

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How Not to Be Wrong summary

Name: How Not to Be Wrong
Rating: 4.1 (112 reviews)
Author: Jordan Ellenberg

Jordan Ellenberg

The Hidden Maths of Everyday Life

4.1 (112 ratings)

20 mins

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How Not to Be Wrong
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What’s in it for me? Learn how math influences everything and how it can help you not be wrong.

Do you find mathematics challenging, or perhaps irrelevant to what you do every day? Many people think math is best left in a classroom, but in reality, it touches our daily lives in profound ways.

That’s because mathematics is really the science of common sense, a reflection of things we already know intuitively.

Mathematicians speak in a specialized language so they can convey complex ideas quickly and precisely. To some, this may make mathematics seem complicated or beyond comprehension.

The mental work involved in complex mathematics, however, isn’t that different from the thinking we do in many common situations. In these blinks, you’ll learn how to uncover the “hidden math” in your life, use it to make sense of problems and importantly, learn not to be wrong.

In the following blinks, you’ll also discover:

why so many research findings published in journals are actually wrong;
why a debut novelist’s second book is usually worse than his best-selling first; and
why the concept of “public opinion” doesn’t actually exist.

Mathematics is the science of not being wrong, and it's based on common sense.

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.

Linearity allows us to simplify mathematical problems.

A basic rule of mathematics is thus: if you have a difficult problem, try to make it easier and solve the easier problem instead. Then hope the simpler version is close enough to the difficult version.

We can break down hard problems into easier ones by assuming they have linearity. In geometry, we assume things have linearity when we consider straight lines. Curves represent nonlinearity.

Drawing conclusions from observational data is questionable, but probability theory can help.

Scientists collect data through observation then use it to build theories. However, observational data can come about by chance, so drawing conclusions from it can be quite precarious.

Consider this example: In 2009, a neuroscientist showed photos of people to a dead fish and measured the fish’s brain activity. Interestingly, the fish responded accurately to the emotions of people in the pictures.

Probability theory tells us what to expect from a bet, but we still have to consider the risks.

In any situation where we're uncertain about an outcome, probability can help us. Probability can't tell us exactly what will happen in the future, but it can tell us what we should expect.

For example, probability theory can tell us what's likely to happen if we place a bet on something. We can use probability to know what might happen when we buy a lottery ticket, by determining the ticket’s expected value.

The regression effect can be found everywhere, but often it isn’t recognized.

Why is a novelist’s second book usually not as good as his first breakout success? Artistic success is subject to a mathematical phenomenon called regression to the mean, or the regression effect.

The regression effect states that if a variable produces an unlikely outcome, the next outcome will tend to be closer to the mean.

Linear regression is useful, but assuming linearity when it isn't there can lead to false conclusions.

As we've seen, linear regression is an important statistical tool that helps us understand how variables relate to each other. However, linear regression can't be used for every set of data; if used incorrectly, it produces misleading results.

You can find the linear regression of a data set by plotting all its points on a graph, then finding the line that comes closest to passing through all of them. This is only meaningful, however, if the data points are already in a generally linear shape.

Many research findings are wrong because of misused data or incorrect probability calculations.

In 2005, a professor named John Ioannidis published a paper called, “Why Most Published Research Findings are False.” That might seem like a radical claim, but his points were sound.

He first stated that insignificant observations can sometimes pass a statistical significance test by chance.

Polls and elections that make statements about “public opinion” are often incorrect.

How can you measure “public opinion”? It’s a rather vague term, so we should be critical when presented with polls that claim to express it.

First, people can have very contradictory opinions, and they can even contradict themselves.

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What is How Not to Be Wrong about?

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.

Best quote from How Not to Be Wrong

Mathematics is the extension of common sense by other means.

—Jordan Ellenberg

Who should read How Not to Be Wrong?

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

About the Author

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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