[Special credit to Ralphjunior for coming up with this fun topic!]
OK, so my math studying days are way, way behind me. Still, when a couple writers on the Craigslist bulletin board started joking about the central paradox of Carly Simon's 1970s hit song "You're so Vain," I couldn't help but feel an urge to dust off the old Set Theory textbook and write this essay.
For those who missed it, Carly Simon recently participated in a charity auction. The item up for sale was the secret identity of the person described in her 1972 song, "You're So Vain." Ever since the song became a No. 1 hit, fans (and even non-fans, like me) have speculated as to who, exactly, Carly was singing about in the song's puzzling chorus:
You're so vain.
You probably think this song is about you...
Adding to the mystery has been Carly's unwillingness to help people decipher the song's lyrics. Various interviewers have tried to pin her down over the years but to no avail: Was it Mick Jagger, James Taylor, Warren Beatty or some other ex-boyfriend? Simon has continually played coy, calling the song a "riddle" to which she refuses to give the answer.
Until this summer, at least. Earlier this month, NBC executive Dick Ebersol plunked down $50,000 to have Simon herself whisper the secret into his ear during lunch at her Massachusetts home. According to various news accounts, one of the conditions of the exchange is that Ebersol has to swear not to divulge the information to anybody else, ever.
In the aforementioned discussion on Craigslist, a number of writers tried to crack the song's central puzzle without assistance from Simon or Ebersol. These attempts inevitably ran smack up against the seemingly self-contradictory nature of song's chorus, which, like the fictional record "I Can't Be Played on Record Player X" in Douglas Hoftstadter's Godel, Escher, Bach, seems to cloak a vexatious paradox that destroys any attempt to logically analyze it.
As Ralphjunior, the poster who launched the thread, succinctly noted:
This song Is about him, he is not vain.
As absurd as the $50,000 auction may have been, it puts a clear monetary value on the solution to this paradox, which I shall hereforth call "Carly's Conundrum." Seeing as how mathematicians love to attach cash prizes to the great unsolved problems of mathematics, I propose adding Carly's Conundrum to this lengthy list. While I can't offer $50,000 like Ebersol, I can promise the satisfaction of undermining the artificially inflated value of this particular piece of intellectual property and sending Ms. Simon back into historical obscurity. For those with less hostile motives, consider this incentive: You will be laying the ground work for solving even more perplexing pop music questions such as:
Who put the Bop in the bop-shoo-bop-shoo-bop?
So what are the ground rules? Well, let's start by assuming that the lyrics of the song represent an internally consistent system S, and go from there. Respondents can either prove or disprove the existence of a particular person p who satisfies the song's definition of who, exactly, "you" may be.
To give an idea of how such a proof might look, I have made the following attempt to disprove the existence of p, using the classic logician's technique of reductio ad absurdum. Like I warned above, my math is rusty, and there are a few logical leaps (and gaps) that the professional mathematician might chuckle at. For the moment, however, I hope this provides a starting point for future inquiry.
Before we get into the proof, I've offered a quick key for the non-mathematicians out there in the audience.
Symbol
Name
Æ
the empty set
Î
element of
Ï
not an element of
Î~
probably an element of
Ï~
probably not an element of
Ç
intersection
È
union
==
Equals
Let p == variable "You"
Carly's Definition 1: p == so vain
Carly's Definition 2: p == probably thinks song S is about p
1. Let A == Set of all very vain people.
{Warren Beatty, Mick Jagger, James Taylor, Donald Trump...}
2 Let B == Set of all people who probably think song S is about them.
{Warren Beatty, James Taylor...}
3. p Î AÈB
[Definitions 1 & 2]
4. pÎ~ B
[Definition 2]
5. p Ï~ A - AÇB
[Converse of 4]
6. |A| >> |B|
[Ecclesiastes et al.]
which means we can safely assume...
7. |A - AÇB| >> |AÇB|
and...
8. p Î~ A-AÇB
[definition of "probably"]
And yet...
9. p Ï~ A- AÇB
[See No. 5]
which leads us to the following awkward statement...
10. For all very vain people p, there exists a sizable proportion for which the reflexive defintion p==p does not hold true.