The Pirahã are a tiny Amazonian tribe, famous for the fact that there number system has three numbers: one, two, and many. This is seen as proof of the Sapir-Whorf hypothesis, that words determine the concepts you can use: Pirahã can’t instantly compare collections of objects to see which is bigger (e.g. a pile of four batteries next to a pile of five) while people in cultures with other number systems can. On the other hand, it’s also seen as a refutation of Sapir-Whorf, on the grounds that the subsistence hunter-gatherers never really need to count, so the words clearly follow the concept rather than the other way around.
I don’t know for sure, but I do know that the Pirahã are nearly correct, just off by one. To a first approximation, for anything you can name, there’s either zero of it, exactly one, or pretty much infinitely many. There’s one me, only one you. There’s a pretty much unlimited supply of people — if you counted seven billion humans, one per second, without stopping to eat or sleep, it would take you 222 years. There are, for almost everyone’s purposes, infinitely many human beings. But most of the possible subsets of humans are empty: ten-foot-tall humans, humans who have lived on Mars, etc.
I didn’t think about this because I was thinking about linguistics. I thought of this because I was waiting for a database query to finish. It occured to me that there were really only three speeds at which computers operate: instantaneously, long enough to get bored and alt-tab, or so long that it was clear you’d done something wrong.
(As it turns out, what was supposed to be an inner join was, due to a funny bug on my part, actually a Cartesian product that would take up like a terabyte of space. You’re welcome, AWS.)
This applies to time-management, too. A mental to-do list has three kinds of items: the thing I am doing now. The thing I will do next. And the things I will never actually do.
Lifestyle changes follow the same pattern. If you’re sad now, the way to get happy is either a) change nothing, because whatever’s making you depressed is not something you have any control over, b) change exactly one thing, or c) change everything. A lot of people tend to assume it’s b when it’s really c, so they make one big life change and end up hating Austin exactly as much as they hated San Francisco, or find a brand new partner to have the same dysfunctional relationship with.
You can trivially construct counterexamples of sets with complicated descriptors and a small-but-greater-than-one number of members, but they’re mostly pedantic. For example, there are three human beings who are descended from me and my wife, and live in Brooklyn. But if you consider all the triples of parent X, parent Y, and city Z, almost all of them are empty, and the modal size of the non-empty sets is probably one.
I think the right way to look at this is as a sort of Platonism. Everything you observe is either a) a fundamental phenomenon written into the universe’s config file, or b) an output of one or more of those phenomena. If it’s a universal phenomenon, the outputs are uncountable.
I’m not the first person to notice this. Ian Fleming has a slightly modified version: “Once is happenstance. Twice is coincidence. Three times is enemy action.” (Maybe he was Pirahã!)
It usually sounds like a silly toy. Think in terms of set theory, find weird conclusions and paradoxes everywhere! Duh.
But it is useful: I wrote most of this post and wasn’t sure where I’d go with it, then I bumped into a friend and asked him what he was working on. He’s doing research into transmissible cancers. Most cancers are not transmissible, but there are a few examples: dogs, hamsters, Tasmanian Devils (transmissible cancers have killed most of them), and clams.
What do these species have in common?
There are two good answers: everything, or nothing. They have everything in common in the sense that they’re living things that are susceptible to runaway cell growth, but that describes almost all complex organisms. There isn’t some common factor between them that isn’t shared by lots of other species.
Which means it’s incredibly improbable that these are the only species with transmissible cancers. Probably, they’re everywhere, but we haven’t spotted them yet. Maybe the Tasmanian Devils are a clue; most of them have been wiped out by transmissible cancer in the last few decades. So one possibility is that transmissible cancers aren’t especially rare historically, but they mostly render their hosts extinct and don’t show up in the fossil record.
So this turns into a fun research problem. Find some phenomenon that happened just a handful of times, and figure out what the underlying cause is, and that cause is probably something close to a law of nature.
 Asking about this distribution might be a pretty good Fermi Estimate question for job interviews. For example, other than paired articles of clothing, are there any distributions like this you can describe where the most common values aren’t zero, followed by one?