How Not to Be Wrong: The Hidden Maths of Everyday Life (Ellenberg, Jordan)
Notes from relevant books on Foreign Policy, Diplomacy, Defence, Development and Humanitarian Action.
Ellenberg, Jordan. How Not to Be Wrong: The Hidden Maths of Everyday Life. Penguin Books Ltd, 2014.
These are my personal notes from this book. They try to give a general idea of its content, but do not in any case replace reading the actual book. Think of them as teasers to encourage you to read further!
When Am I Going to Use This?
When American planes came back from engagements over Europe, they were covered in bullet holes. But the damage wasn’t uniformly distributed across the aircraft. There were more bullet holes in the fuselage, not so many in the engines.
The officers saw an opportunity for efficiency; you can get the same protection with less armor if you concentrate the armor on the places with the greatest need, where the planes are getting hit the most.
The armor, said Wald, doesn’t go where the bullet holes are. It goes where bullet holes aren’t: on the engines.
The reason planes were coming back with fewer hits to the engine is that planes that got hit in the engine weren’t coming back.
Here’s an old mathematician’s trick that makes the picture perfectly clear: set some variables to zero.
A mathematician is always asking, “What assumptions are you making? And are they justified?”
The structure underlying the bullet hole problem is a phenomenon called survivorship bias. It arises again and again, in all kinds of contexts.
But something’s missing: the funds that aren’t there. Mutual funds don’t live forever. Some flourish, some die. The ones that die are, by and large, the ones that don’t make money.
MATHEMATICS IS THE EXTENSION OF COMMON SENSE BY OTHER MEANS
Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.
Part I: Linearity.
One: Less Like Sweden
Why is Obama trying to make America more like Sweden when Swedes are trying to be less like Sweden?
difference between linearity and nonlinearity, one of the central distinctions in mathematics.
Nonlinear thinking means which way you should go depends on where you already are.
Two: Straight Locally, Curved Globally
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.
Four: How Much is That in Dead Americans?
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.
Five: More Pie Than Plate
Don’t talk about percentages of numbers when the numbers might be negative.
Six: The Baltimore Stockbroker and the Bible Code
When a company launches a mutual fund, they often maintain the fund in-house for some time before opening it to the public, a practice called incubation.
Improbable things happen a lot.
Seven: Dead Fish Don’t Read Minds
It’s not enough that the data be consistent with your theory; they have to be inconsistent with the negation of your theory, the dreaded null hypothesis.
Fourteen: The Triumph of Mediocrity
With time, the top performers started to look and behave just like the members of the common
A phenomenon now called regression to the mean.
false linearity.
The triumph of mediocrity observed by Secrist, Hotelling points out, is more or less automatic whenever we study a variable that’s affected by both stable factors and the influence of chance.
“The thesis of the book,” he writes in response, “when correctly interpreted, is essentially trivial. … To ‘prove’ such a mathematical result by a costly and prolonged numerical study of many kinds of business profit and expense ratios is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoölogy or mathematics.”
Fifteen: Galton‘s Ellipse
The late nineteenth century was a kind of golden age of data visualization. In 1869 Charles Minard made his famous chart showing the dwindling of Napoleon’s army on its path into Russia and its subsequent retreat, often called the greatest data graphic ever made; this, in turn, was a descendant of Florence Nightingale’s cox-comb graph showing in stark visual terms that most of the British soldiers lost in the Crimean War had been killed by infections, not Russians.
“correlation does not imply causation”—two phenomena can be correlated, in Galton’s sense, even if one doesn’t cause the other.
Darwin showed that one could meaningfully talk about progress without any need to invoke purpose. Galton showed that one could meaningfully talk about association without any need to invoke underlying cause.
it’s common practice among mathematicians and other trig aficionados to use the word “orthogonal” to refer to something unrelated to the issue at hand—“ You might expect that mathematical skills are associated with magnificent popularity, but in my experience, the two are orthogonal.”
The story is more complicated than the Brooksian portrait of a new breed of latte-sipping, Prius-driving liberals with big tasteful houses and NPR tote bags full of cash. In fact, rich people are still more likely to vote Republican than poor people are, an effect that’s been consistently present for decades.
As the voters get more informed, they don’t get more Democratic or more Republican, but they do get more polarized: lefties go farther left, right-wingers get farther right, and the sparsely populated space in the middle gets even sparser. In the lower half of the graph, the less-informed voters tend to adopt a more centrist stance.
They’re undecided because they’re barely paying attention.
Sixteen: Does Lung Cancer Make You Smoke Cigarettes?
surrogate endpoint problem.
Seventeen: There is No such Thing as Public Opinion
“The most plausible reading of this data is that the public wants a free lunch,” economist Bryan Caplan wrote. “They hope to spend less on government without touching any of its main functions.”
Scalia’s mistake is the same one that constantly trips up attempts to make sense of public opinion; the inconsistency of aggregate judgments.
The mathematical buzzword in play here is “independence of irrelevant alternatives.”
What bothers us about Al Gore’s loss in Florida? It’s that more people preferred Gore to Bush than the reverse. Why doesn’t our voting system know
But if you introduce a really small unlit pile of oats, the small dark pile looks better by comparison; so much so that the slime mold decides to choose it over the big bright pile almost all the time. This phenomenon is called the “asymmetric domination effect,”
Chris was the irrelevant alternative; an option that was plainly worse than one of the choices already on offer. You can guess what happened. The presence of a slightly dumber version of Adam made the real Adam look better;
Eighteen: “Out of Nothing I have Created a Strange New Universe”
What if the parallel axiom were false? Does a contradiction follow? And he found that the answer was no—that there was another geometry, not Euclid’s but something else, in which the first four axioms were correct but the parallel postulate was not.
shore up the logical underpinnings of mathematics.
ouroboric, after the mythical snake so hungry it chows down on its own tail and consumes itself.
Mark Twain is good on this: “It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a telephone or any other important thing—and the last man gets the credit and we forget the others.”
Gödel, whose theorem ruled out the possibility of definitively banishing contradiction from arithmetic, was also worried about the Constitution, which he was studying in preparation for his 1948 U.S. citizenship test. In his view, the document contained a contradiction that could allow a Fascist dictatorship to take over the country in a perfectly constitutional manner.
How to be Right
Theodore Roosevelt, from his speech “Citizenship in a Republic,” delivered in Paris in 1910, shortly after the end of his presidency: It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.
Roosevelt returns throughout the speech, is that the survival of civilization depends on the triumph of the bold, commonsensical, and virile against the soft, intellectual, and infertile.
Against Roosevelt I place John Ashbery, whose poem “Soonest Mended”
Philosophers of a mathematical bent call this brittleness in formal logic ex falso quodlibet, or, among friends, “the principle of explosion.”
Samuel Beckett later put it more succinctly: “I can’t go on, I’ll go on.”
It’s not clear how much higher math Beckett knew, but in his late prose piece Worstward Ho, he sums up the value of failure in mathematical creation more succinctly than any professor ever has: Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.