How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

Fortunately, there’s a better answer. It goes something like this: “Mathematics is not just a sequence of computations to be carried out by rote until your patience or stamina runs out—although it might seem that way from what you’ve been taught in courses called mathematics. Those integrals are to mathematics as weight training and calisthenics are to soccer. If you want to play soccer—I mean, really play, at a competitive level—you’ve got to do a lot of boring, repetitive, apparently pointless drills. Do professional players ever use those drills? Well, you won’t see anybody on the field curling a weight or zigzagging between traffic cones. But you do see players using the strength, speed, insight, and flexibility they built up by doing those drills, week after tedious week. Learning those drills is part of learning soccer.

If you go to the recovery room at the hospital, you’ll see a lot more people with bullet holes in their legs than people with bullet holes in their chests. But that’s not because people don’t get shot in the chest; it’s because the people who get shot in the chest don’t recover.

A mathematician is always asking, “What assumptions are you making? And are they justified?” This

Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

You can’t do calculus by common sense. But calculus is still derived from our common sense—Newton

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.

Without the rigorous structure that math provides, common sense can lead you astray.

our ignorance is much greater than our knowledge.

Nonlinear thinking means which way you should go depends on where you already are.

If you have before you a square whose side has length X, its area is X times X—indeed, that’s why we call the operation of multiplying a number by itself squaring!

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.

The trouble started with Zeno, a fifth-century-BCE Greek philosopher of the Eleatic school who specialized in asking innocent-seeming questions about the physical world that inevitably blossomed into huge philosophical brouhahas.

Cauchy did not comply. Cauchy was not interested in the needs of engineers. Cauchy was interested in the truth.

I have some good news. We’re not all going to be overweight in the year 2048. Why? Because not every curve is a line.

Calculators are also useful tools that people worked hard to make—we should use them, too, when the situation demands!

If we settle on a vision of mathematics that consists of “getting the answer right” and no more, and test for that, we run the risk of creating students who test very well but know no mathematics at all.

That’s how the Law of Large Numbers works: not by balancing out what’s already happened, but by diluting what’s already happened with new data, until the past is so proportionally negligible that it can safely be forgotten.

Most mathematicians would say that, in the end, the disasters and atrocities of history form what we call a partially ordered set. That’s a fancy way of saying that some pairs of disasters can be meaningfully compared and others cannot.

Don’t talk about percentages of numbers when the numbers might be negative.

The combination of positive and negative allows you, if you’re not careful, to tell a fake story,

Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.

This vision of religious belief is extremely congenial to the mathematical mind. You believe in God not because you were touched by an angel, not because your heart opened up one day and let the sunshine in, and certainly not because of something your parents told you, but because God is a thing that must be, as surely as 8 times 6 must be the same as 6 times 8.

This didn’t faze Tversky, who relished a good fight, whatever the outcome. “I’ve been in a thousand arguments over this topic,” he said. “I’ve won them all, and I’ve convinced no one.”

In part, the replicability crisis is simply a reflection of the fact that science is hard and that most ideas we have are wrong—even most of those ideas that survive a first round of prodding.

But a conventional boundary, obeyed long enough, can be easily mistaken for an actual thing in the world.

Data is messy, and inference is hard.

The purpose of a court is not truth, but justice. We have rules, the rules must be obeyed, and when we say that a defendant is “guilty” we mean, if we are careful about our words, not that he committed the crime he’s accused of, but that he was convicted fair and square according to those rules. Whatever rules we choose, we’re going to let some criminals go free and imprison some of the blameless.

The significance test is the detective, not the judge.

The age of Big Data is frightening to a lot of people, and it’s frightening in part because of the implicit promise that algorithms, sufficiently supplied with data, are better at inference than we are.

In the Bayesian framework, how much you believe something after you see the evidence depends not just on what the evidence shows, but on how much you believed it to begin with.

If you do happen to find yourself partially believing a crazy theory, don’t worry—probably the evidence you encounter will be inconsistent with it, driving down your degree of belief in the craziness until your beliefs come into line with everyone else’s. Unless, that is, the crazy theory is designed to survive this winnowing process. That’s how conspiracy theories work.

The lesson about inference: you have to be careful about the universe of theories you consider. Just as there may be more than one solution to a quadratic equation, there may be multiple theories that give rise to the same observation, and if we don’t consider them all, our inferences may lead us badly astray.

This is the phenomenon of perspective; when you try to depict the three-dimensional world on your two-dimensional field of vision, something has to give.

That’s part of the glory of math; we develop a body of ideas, and once they’re correct, they’re correct, even when applied far, far outside the context in which they were first conceived.

Shannon, in the paper that launched the theory of information, identified the basic tradeoff that engineers still grapple with today: the more resistant to noise you want your signal to be, the slower your bits are transmitted.

That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.

it’s hard to separate our moral feelings about an activity from the judgments we make about its rationality.

You can replace your name by an alias, but there’s no alias for the shape of your head.

like those pious people who, over time, accumulate a sense of their own virtuousness so powerful as to make them believe the bad things they do are virtuous too.

The ballot down under looks a lot like Borda’s. You don’t just mark your ballot with the candidate you like best; you rank all the candidates, from your favorite to the one you hate the most.

Instant-runoff voting (IRV) has been around for more than 150 years. They use it not only in Australia but in Ireland and Papua New Guinea.

It’s not wrong to say Hilbert was a genius. But it’s more right to say that what Hilbert accomplished was genius. Genius is a thing that happens, not a kind of person.

But we have committed ourselves to Condorcet’s more fundamental belief, that a quantitative “social mathematics”—what we now call “social science”—ought to have a part in determining the proper conduct of government.

When you reason correctly, as Silver does, you find that you always think you’re right, but you don’t think you’re always right. As the philosopher W. V. O. 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.”

We are tolerant of contradiction, to 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.”

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

When are you going to use it? You’ve been using mathematics since you were born and you’ll probably never stop. Use it well.