All Real Paradoxes Are About Self-Reference
I’ve been thinking about what makes a paradox really paradoxical. My hunch is that all genuine paradoxes are just different guises—homeomorphisms, if you will—of problems involving recursion and self-reference. They’re systems trying to step outside themselves, and failing. You can think of them as homeomorphisms of recursion.
Yes, I’m using the word homeomorphism partly to sound clever after finishing topology. But I’m also using it express that these paradoxes share an underlying structure. They may look different on the surface, but you can deform one into another without tearing the logic. They're all twisted versions of the same core problem.
The cleanest example is the liar paradox:
“This proposition is false.”
If it's true, it's false. If it's false, it's true. It loops back on itself like a snake eating its tail. The statement tries to evaluate itself from within the same system it belongs to and breaks the system in doing so.
Another case is the Pop Quiz Paradox:
A teacher announces there will be a surprise quiz next week. The students reason that it can’t be on Friday, because they’d know by Thursday night if they had yet to be quizzed that the quiz would be the next day. They cannot be surprised. If it can’t be on Friday, it can’t be on Thursday by the same logic. Then it can’t be Wednesday or Tuesday, or Monday, so they reason it can’t happen at all. Yet when the quiz comes (on Wednesday, e.g.) they are all genuinely surprised. Again, this is a statement being made within a system: surprisal in the week from within the lived experience of the week. That is where the paradox arises.
Then there are what I would argue are false paradoxes, like Zeno’s Paradox: Achilles gives a tortoise a head start in a race. To catch it, he must first reach the point where the tortoise began. By then, the tortoise has moved ahead. He must then reach that new point, and so on infinitely. So, it seems Achilles can never catch the tortoise. This so called “paradox” that does not have to do with self-reference vanishes once you understand calculus. The Ancient Greeks just had a lack of tools.
The true paradoxes, the ones that linger and never resolve, all seem to emerge from the same strange place: systems that try to describe themselves. They are strange loops (read Douglas Hofstadter!) that press against their own boundaries and collapse inward. Recursion is the heart of the mystery.