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August 27, 2018 - November 14, 2020
But what kinds of things? Give me
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.
Nonlinear thinking means which way you should go depends on where you already are.
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.
(In mathematics, you very seldom get the clearest account of an idea from the person who invented it.)
which for present purposes is
convert, and the more slowly the
In fact, if you flip enough coins, there’s only the barest chance of getting as many as 51%! Observing a highly unbalanced result in ten flips is unremarkable; getting the same proportional imbalance in a hundred flips would be so startling as to make you wonder whether someone has mucked with your coins.
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.
What does this mean for you, if you’re fortunate enough to have some money to invest? It means you’re best off resisting the lure of the hot new fund that made 10% over the last twelve months. Better to follow the deeply unsexy advice you’re probably sick of hearing, the “eat your vegetables and take the stairs” of financial planning: instead of hunting for a magic system or an adviser with a golden touch, put your money in a big dull low-fee index fund and forget about it.
Awesome
When you’re trying to draw reliable inferences from improbable events, wiggle room is the enemy.
I once had an uneasy moment with a colleague in an airport when he made the remark, unexceptional in a mathematical context, that it might be necessary to blow up the plane at one point.
If only we could go back in time to the dawn of statistical nomenclature and declare that a result passing Fisher’s test with a p-value of less than 0.05 was “statistically noticeable” or “statistically detectable” instead of “statistically significant”! That would be truer to the meaning of the method, which merely counsels us about the existence of an effect but is silent about its size or importance. But it’s too late for that. We have the language we have.* THE
But here’s the bad news: the reductio ad unlikely, unlike its Aristotelian ancestor, is not logically sound in general. It leads us into its own absurdities. Joseph Berkson, the longtime head of the medical statistics division at the Mayo Clinic, who cultivated (and loudly broadcast) a vigorous skepticism about methodology he thought shaky, offered a famous example demonstrating the pitfalls of the method.
This is a test
we think of primes as random numbers. The reason the fruitfulness of this viewpoint is so remarkable is that the viewpoint is so very, very false. Primes are not random! Nothing about them is arbitrary or subject to chance. Quite the opposite: we take them as immutable features of the universe, and carve them on the golden records we shoot out into interstellar space to prove to the ETs that we’re no dopes.
But noise is just as likely to push you in the opposite direction from the real effect as it is to tell the truth. So we’re left in the dark by a result that offers plenty of statistical significance but very little confidence.
“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. You
“in fact no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas.”
they did. Why would they? Where’s the money in it? Maybe that’s right.
totally ignores the proportion of
That’s not how the Iranian vote counts looked. There were too many 7s, almost twice as many as their fair share; not like digits derived from a random process, but very much like digits written down by humans trying to make them look random. This, by itself, isn’t proof that the election was fixed, but it’s evidence in that direction.
a bigger payoff the more numbers a
Halley was scientist enough to understand the absurdity of the age-independent pricing scheme. He determined to work out a more rational accounting of the value of a lifetime annuity.
This reminds me of an old story from the Harvard math department, concerning one of the grand old Russian professors, whom we shall call O. Professor O is midway through an intricate algebraic derivation when a student in the back row raises his hand. “Professor O, I didn’t follow that last step. Why do those two operators commute?” The professor raises his eyebrows and says, “Eet ees obvious.” But the student persists: “I’m sorry, Professor O, I really don’t see it.” So Professor O goes back to the board and adds a few lines of explanation. “What we must do? Well, the two operators are both
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“Nothing seems to happen, and yet at the end a highly nontrivial theorem is there.”
If anything, I’ve found that in moments of emotional extremity there is nothing like a math problem to quiet the complaints the rest of the psyche serves up.
“If you never miss the plane, you’re spending too much time in airports.”
The younger Bernoulli’s beautiful untwisting of the paradox is a landmark result, and one that has formed the foundation of economic thinking about uncertain values ever since.
Mankiw was already at an equilibrium, where the utility of the dollars he’d earn from another hour of work would be exactly canceled by the negative utility imposed by the loss of an hour with his kids.
rich states vote for Democrats but rich people vote for Republicans,
By means of the eccentricity—a geometric quantity at least as old as the work of Apollonius of Perga in the third century BCE—Galton had found a way to measure the association between two variables, and in so doing had solved a problem at the cutting edge of nineteenth-century biology: the quantification of heredity.
It’s been a while since a mathy word really broke out into demotic English. Lowest common denominator has by now lost its mathematical flavor almost entirely, and exponentially—just don’t get me started on exponentially.*
The human body is an immensely complex system, and there are only a few of its features we can measure, let alone manipulate. Based on the correlations we can observe, there are lots of drugs that might plausibly have a desired health effect. And so you try them out in experiments, and most of them fail dismally. To work in drug development requires a resilient psyche, not to mention a vast pool of capital.
Keep this in mind when you’re told that two phenomena in nature or society were found to be uncorrelated. It doesn’t mean there’s no relationship, only that there’s no relationship of the sort that correlation is designed to detect.
But the correlation between HDL and absence of heart attack might not indicate any causal link.
refraining from making recommendations at all, on the grounds that they might be wrong, is a losing strategy.
“Statistics can soundly play an ancillary role in its elucidation. But if biologists permit statisticians to become arbiters of biologic questions, scientific disaster is inevitable.”
So what looks at first like a causal relationship between high blood pressure and diabetes is really just a statistical phantom.
want spending cut, but are opposed to cuts in anything except foreign aid. . . . The conclusion is inescapable: Republicans have a mandate to repeal the laws of arithmetic.”
How are you supposed to know what the public wants when the public makes no sense?
When the House of Representatives debated adoption of the Bill of Rights in August 1789, Samuel Livermore of New Hampshire argued that the vagueness of the language would allow softhearted future generations to outlaw necessary punishments:
An election is a machine. I like to think of it as a big cast-iron meat grinder. What goes into the machine is the preferences of the individual voters. The sausagey goop that comes out, once you turn the crank, is what we call the popular will.
So if you’re a single guy looking for love, and you’re deciding which friend to bring out on the town with you, choose the one who’s pretty much exactly like you—only slightly less desirable.
This is not just an argument about politics—it’s a fundamental question that applies to every field of mental endeavor. Are we trying to figure out what’s true, or are we trying to figure out what conclusions are licensed by our rules and procedures? Hopefully the two notions frequently agree; but all the difficulty, and thus all the conceptually interesting stuff, happens at the points where they diverge.
“Long live formalism. It is what makes a government a government of laws and not of men.”
Formalism’s greatest champion in mathematics was David Hilbert, the German mathematician whose list of twenty-three problems, prepared for a lecture in Paris at the 1900 International Congress of Mathematics, set the course for much of twentieth-century math. Hilbert is so revered that any work that touches even tangentially on one of his problems takes on a little extra shine, even a hundred years later. I
“One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.”
“The typical working mathematician is a Platonist on weekdays and a formalist on Sundays.”
You’ve been using mathematics since you were born and you’ll probably never stop. Use it well.
