Q: What is a self-reference?
A: It’s a statement that refers to itself.
Q: So, like… is this question self-referential?
Q: And this question too?!
A: You can stop now.
Q: So why is self-reference important in mathematics?
A: Because mathematics is built on logic, and self-references unravel logic.
Q: How so?
A: Allow me: “This sentence is false.”
A: Now, is that statement true or false?
Q: False, obviously. It says so.
A: Are you sure? If it’s false, then we have to reject what it says… meaning that it’s true.
Q: Weird! Okay, so it’s true.
A: Ah, but if it’s true, then we have to accept what it says… meaning that it’s false.
Q: So if it’s false, then it’s true, and if it’s true, then it’s false?
A: Self-reference: frying brains since 1901.
Q: All right, good riddle. But what does this have to do with math?
A: I’m glad you asked!
Q: You’re “glad”? You realize we’re being written by the same person, right?
A: Whoa, whoa, too much self-reference there, pal.
Q: Anyway. You were saying?
A: Here’s what mathematicians do: they make some assumptions, and then, working logically from those assumptions, they prove as much as they can.
It’s called “deductive reasoning.” Highly recommended.
In any case, mathematicians want to believe that, if they pick the right assumptions, they can prove everything. Every statement will be either true or false, with no annoying gray areas.
Q: What does this have to do with self-reference?
A: Self-reference creates annoying gray areas.
Q: Interesting! Go on.
A: Hey, I’ll go on when I feel like it! You don’t tell me what to do.
Q: Dude. C’mon.
A: Okay! So here’s how self-reference creates annoying gray areas in math.
Every system of assumptions creates a kind of language. To be clear, it’s a super technical language. Not fun to speak. No silly accents. But it’s a language nevertheless, with all the associated logical structure and expressive variety.
Now, you can use this language to create self-references. In particular, in this language, you can make a sentence which says, “This statement cannot be proved true.”
Q: Uh-oh. That sounds bad.
A: It is! Because, if the statement is false…
Q: Oh, I see! If the statement is false, then we have to reject what it says, meaning that it can be proved true… but that means it’s a false statement that can be proved true!
A: Right. Contradiction.
Q: So this statement can’t be false. And it can’t be true either… because… why?
A: Oh, this statement actually can be true.
Q: It can?
A: Yes. In fact, it is.
Q: What?! You said there were gray areas!
A: There are. The statement is true, but—here’s the gray area—it can’t be proved true.
A: Think about it. We’re assuming this statement is true. But if we prove it true…
Q: Then we’ve contradicted it. That’s messed up.
Q: So what’s the deal with this statement?
A: Like I said, it’s true, but it cannot be proved true.
Q: Okay… but isn’t the argument you just gave me a proof that the statement is true?
A: Not really. “Provable” in this case means “follows logically from the assumptions.” I haven’t accomplished that. I’m not working inside the system. Instead, I’m critiquing and dissecting it from the outside.
Q: This is madness!
A: No, this is logic!
What we’re talking about is Gödel’s Incompleteness Theorem. It’s one of the most baffling and beautiful results in all of modern mathematics.
A: Logic can be, yes.
What this result says is that mathematics can never be complete. No matter what assumptions you make, there will always be self-referential statements beyond the reach of proof.
Q: Honestly, when I started asking about self-reference, I wasn’t expecting any big insights about the nature of mathematical truth. I just wanted to make some dumb jokes.
A: Like how “hippopotomonstrosesquipedaliophobia” means “fear of long words”?
A: Or how certain adjectives are self-describing, like “unhyphenated”?
Q: Or “self-describing”!
A: Or “containing an even number of syllables.”
Q: Or “containing an odd number of syllables.”
A: Yeah, but that one’s just a copy of mine.
Q: It’s not a copy! It’s an homage.
A: Hmmph. It’s less an homage than a reference.
Q: A self-reference?
A: I’m leaving.