How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

One simple and popular strategy is to report confidence intervals in addition to p-values.

For Neyman and Pearson, the purpose of statistics isn’t to tell us what to believe, but to tell us what to do. Statistics is about making decisions, not answering questions. A significance test is no more or less than a rule, which tells the people in charge whether to approve a drug, undertake a proposed economic reform, or start up a website.

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. You know how when you read an article about a breakthrough finding that this thing causes that thing, or that thing prevents the other thing, and at the end there’s always a banal sort of quote from a senior scientist not involved in the study intoning some very minor variant of “The finding is quite interesting, and suggests that more research in this direction is needed”? And how you don’t really even read that part because you think of it as an obligatory warning without content? Here’s the thing—the reason scientists always say that is because it’s important and it’s true!

Relying purely on null hypothesis significance testing is a deeply non-Bayesian thing to do—strictly speaking, it asks us to treat the cancer drug and the plastic Stonehenge with exactly the same respect.

“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth, unless the truth is a hypothesis it didn’t occur to you to consider.”

What about GODS, where the world was put together in a hurry by a squabbling committee? Many distinguished civilizations have believed as much. And you can’t deny that there are aspects of the natural world—I’m thinking pandas here—that seem more likely to have resulted from grudging bureaucratic compromise than from the mind of an all-knowing deity

Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.

when you win the lottery a lot, the IRS audits you a lot,

What is this, but a reckoning of the costs and benefits of adopting faith? Even in the middle of ecstatic communion with his savior, Pascal was still doing math! I love this about him.

His attitude toward Pascal is that of the popular smart kid toward the bitter and nonconforming nerd.

Pascal is not trying to convince you God exists; he is trying to convince you that it would be to your benefit to believe so,

In the decision-theory literature, the former kind of unknown is called risk, the latter uncertainty. Risky strategies can be analyzed numerically; uncertain strategies, Ellsberg suggested, were beyond the bounds of formal mathematical analysis, or at least beyond the bounds of the flavor of mathematical analysis beloved at RAND.

There’s something the mind resists about regression to the mean. We want to believe in a force that brings down the mighty. It’s not satisfying enough to accept what Galton knew in 1889: the apparently mighty are seldom quite as mighty as they look.

he is trying to explain to a distinguished colleague, in the kindest way possible, that he has wasted ten years of his life.

What can I say? Mathematics is a way not to be wrong, but it isn’t a way not to be wrong about everything. (Sorry, no refunds!) Wrongness is like original sin; we are born to it and it remains always with us, and constant vigilance is necessary if we mean to restrict its sphere of influence over our actions. There is real danger that, by strengthening our abilities to analyze some questions mathematically, we acquire a general confidence in our beliefs, which extends unjustifiably to those things we’re still wrong about. We become 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 graph reflects a sobering social fact, which is by now commonplace in the political science literature. Undecided voters, by and large, aren’t undecided because they’re carefully weighing the merits of each candidate, unprejudiced by political dogma. They’re undecided because they’re barely paying attention.

If you never give advice until you’re sure it’s right, you’re not giving enough advice.

the average American is always eager to slash foreign aid, occasionally tolerant of cuts to welfare or defense, and pretty gung ho for increased spending on every single other program our taxes fund.

Not because the voters are stupid or delusional. Each voter has a perfectly rational, coherent political stance. But in the aggregate, their position is nonsensical.

In other words: the infantile “average American,” who wants to cut spending but demands to keep every single program, doesn’t exist. The average American thinks there are plenty of non-worthwhile federal programs that are wasting our money and is ready and willing to put them on the chopping block to make ends meet. The problem is, there’s no consensus on which programs are the worthless ones.

The “majority rules” system is simple and elegant and feels fair, but it’s at its best when deciding between just two options. Any more than two, and contradictions start to seep into the majority’s preferences.

If there’s no such thing as public opinion, what’s an elected official to do? The simplest answer: when there’s no coherent message from the people, do whatever you want. As we’ve seen, simple logic demands that you’ll sometimes be acting contrary to the will of the majority.

Here you see one of IRV’s weaknesses. A centrist candidate who’s liked pretty well by everyone, but is nobody’s first choice, has a very hard time winning.

Are we, as Fisher thought, trying to figure out which hypotheses we should actually believe are true? Or are we to follow the Neyman-Pearson philosophy, under which we resist thinking about the truth of hypotheses at all and merely ask: Which hypotheses are we to certify as correct, whether they’re really true or not, according to our chosen rules of inference?

Russell’s letter. Despite the respectful tone (“I have encountered a difficulty,” not “Hi, I’ve just borked your life’s work”), Frege understood at once what Russell’s paradox meant for his version of set theory.

In his view, the document contained a contradiction that could allow a Fascist dictatorship to take over the country in a perfectly constitutional manner.

“The closet philosopher, the refined and cultured individual who from his library tells how men ought to be governed under ideal conditions, is of no use in actual governmental work,”

Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying “huh,” but rather making a firm assertion: “I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.” Or even more: “I’m unsure, and you should be too.”

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. Don’t cheat!

What’s true is that the sensation of mathematical understanding—of suddenly knowing what’s going on, with total certainty, all the way to the bottom—is a special thing, attainable in few if any other places in life. You feel you’ve reached into the universe’s guts and put your hand on the wire. It’s hard to describe to people who haven’t experienced it. We are not free to say whatever we like about the wild entities we make up. They require definition, and having been defined, they are no more psychedelic than trees and fish; they are what they are. To do mathematics is to be, at once, touched by fire and bound by reason. This is no contradiction. Logic forms a narrow channel through which intuition flows with vastly augmented force.

“I want to yell at people, at length, about how great math is” to something more like an actual book.