How Not To Be Wrong: The Hidden Maths of Everyday

by Jordan Ellenberg

Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.

A basic rule of mathematical life: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.

Working an integral or performing a linear regression 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. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

It’s easy to think of the quantitative analysis of policy as something you do with a calculator. But the calculator only enters once you’ve figured out what calculation you want to do. I

Impossible things never happen. But improbable things happen a lot.

“A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.” Not “succeeds once in giving,” but “rarely fails to give.” A statistically significant finding gives you a clue, suggesting a promising place to focus your research energy. The significance test is the detective, not the judge.

When you’re faced with a math problem you don’t know how to do, you’ve got two basic options. You can make the problem easier, or you can make it harder. Making it easier sounds better—you replace the problem with a simpler one, solve that, and then hope that the understanding gained by solving the easier problem gives you some insight about the actual problem you’re trying to solve.

Charles Minard made his famous chart showing the dwindling of Napoleon’s army on its path into Russia and its subsequent retreat,

Florence Nightingale’s cox-comb graph†

As the philosopher W. O. V. Quine put it, “To believe something is to believe that it is true; therefore a reasonable person believes each of his beliefs to be true; yet experience has taught him to expect that some of his beliefs, he knows not which, will turn out to be false. A reasonable person believes, in short, that each of his beliefs is true and that some of them are false.”

There’s such a thing as being too precise. The models we use to score standardized tests could give SAT scores to several decimal places, if we let them, but we shouldn’t—students are anxious enough as it is, without having to worry about their classmate nosing ahead of them by a hundredth of a point.

As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposed ideas in the mind at the same time, and still retain the ability to function.”

Proving by day and disproving by night is not just for mathematics. I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical. Believe whatever you believe by day; but at night, argue against the propositions you hold most dear.

Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.

The lessons of mathematics are simple ones and there are no numbers in them: that there is structure in the world; that we can hope to understand some of it and not just gape at what our senses present to us; that our intuition is stronger with a formal exoskeleton than without one. And that mathematical certainty is one thing, the softer convictions we find attached to us in everyday life another, and we should keep track of the difference if we can.