Although I tend to prefer David Foster Wallace’s nonfiction, one of my favorite things I’ve read by him comes from his first novel, The Broom of the System:
“…a second-order vain person is first of all a vain person. He’s vain about his intelligence, and wants people to think he’s smart. Or his appearance, and wants people to think he’s attractive. Or, say, his sense of humor, and wants people to think he’s amusing and witty. Or his talent, and wants people to think he’s talented. Et cetera. You know what a vain person is.”
“Right.”
“A vain person is concerned that people not perceive him as stupid, or dull, or ugly, et cetera et cetera.”
“Gotcha.”
“Now a second-order vain person is a vain person who’s also vain about appearing to have an utter lack of vanity. Who’s enormously afraid that other people will perceive him as vain. A second-order vain person will sit up late learning jokes in order to appear funny and charming, but will deny that he sits up late learning jokes. Or he’ll perhaps even try to give the impression that he doesn’t regard himself as funny at all.”
“…”
“A second-order vain person will be washing his hands in a public restroom, and will be unable to resist the temptation to admire himself in the mirror, to scrutinize himself, but he’ll pretend he’s fixing a contact lens or getting something out of his eye while he does so, so that people won’t perceive him as the sort of person who admires himself in mirrors, but rather as the sort of person who uses mirrors only to attend to reasonable, un-vain business.”
The thing is, if you start off vain you can’t leap straight to being non-vain. Even if you think your intentions are pure, at some level your desire to be selfless is going to be motivated by the fact that selfless people are more likeable. But you can jump from first-order to second-order vanity, and in fact this is usually quite effective in getting people to like you more and getting yourself to do The Right Thing™ (i.e. what a selfless person would do). But it’s not perfect; your vain bias will inevitably seep through sometimes, at which point perceptive people will notice that you’re only selfless to first order.
You can tell DFW’s math background from this quote, because of the fun analogy to Taylor series. For background: the Taylor series of a function $f(x)$ is an “infinite-degree polynomial” in $x$ (an expression like $a_0 + a_1x + a_2x^2 + a_3x^3 + \dots$ continued for all $n$) that can be used to approximate $f$. By truncating the series after the $x^n$ term you obtain the $n$th order Taylor approximation. As you increase the order, the approximation becomes closer and closer to the actual function.
Similarly, I hypothesize that a sufficiently dedicated vain person who wants not to be vain can achieve third- or even fourth-order vanity; that is, concealing not only your vanity, but the fact that your supposed selflessness is in fact motivated by vanity. Each additional layer of concealment causes you to get additional “corner cases” right (i.e. act selfless in more and more situations), but there will always be situations unusual enough to confuse you and make your vanity show through.
Not that such completely selfless people really exist, probably, so maybe we should just try to be higher-order than the next guy?