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Mike Z. writes,
Hey bro, do you mind going over a couple paradoxes? Zeno’s paradoxes, like what’s the deal with them. Trying to explain that shit is fucking hard, think you can do it?
So maybe one of the reasons you’re having trouble explaining paradoxes is that paradoxes are fucking hard to explain, and that’s the whole thing about paradoxes. What is a paradox? Roughly, something that makes us go, “Oh, I get it, just… fuck, that does not make sense.” Sometimes they generate straight-up contradictions - those ones are easy enough to understand, but hard to wrap your whole brain around, which is the point.
Take Russell’s Paradox: Does the set of sets that don’t contain themselves contain itself or not? If it does, then it doesn’t - but if it doesn’t, then it does. We used to think that you could just, like, name a set and you would get a set - like, “Uh, the set of all odd numbers!” and poof there it was. In one sense, Russell’s paradox is a super-clear demonstration that “sorry assholes, it’s not that simple.” But in another sense we’re like, “Wait, so… the set has sets that don’t contain themselves? So it contains itself… which means it can’t contain itself… so it doesn’t… but then it would have to? That can’t be right, can it?”
So Russell developed this paradox and we responded by going, “Oh! Fuck, okay. Well, we have to get rid of that easy way of doing sets.” We got lucky there, because we knew exactly what to throw out and how to start over and now we have set theory. Hooray! But there isn’t always a clear culprit. Zeno’s paradoxes bring that out particularly well.
“Zeno’s Paradoxes” are named for one of several Zenos (Zenoes? Zenos.) from ancient Greece. A super-short version of a really famous one goes something like this: Say Achilles is running to his ship. No reason, just trying to keep in shape because he has a feeling he’s going to be on this beach for a long. fucking. time. Now, to get to the ship he has to run halfway there first. So he gets to halfway, and now he has to cover half of the new distance. So he covers half of that, and now he has to cover half of this new new distance. He keeps getting closer and closer but he only ever gets halfway there, so he never makes it all the way.
It’s like, if you owed me money, and you always repaid only half of what you owe me, you’d never pay off the entire debt. And then when you got to $.01 and started paying me half-pennies and quarter-pennies I’d be like “Fuck you, dude, forget it!” But with distance, according to Zeno, you can’t do that. You have to cover the last bit of distance, but you can’t ever cover the last bit of distance. In one sense, that’s pretty straightforward - half is less than a whole, no teleporting through space. Got it. But it’s confusing because it says you literally can’t go places, and we’ve all been places. I’ve been lots of places, and some of them aren’t even here, where I am now.
So, uh, what do I change to make it make sense? Do I get rid of the part where halfway isn’t the whole way? Maybe if there’s a smallest unit, like a penny but for spacetime, that would work. But we have no idea if that’s true or not.* We know now that if you just keep adding (½ + ¼ + 1/8…) you get 1, but Zeno might be like, “Sure, asshole, after infinitely many steps he’d get there, but how is he supposed to take infinitely many steps in like two minutes?” Aristotle pointed out that as the distance gets smaller, so should time. You can read about the proposed solutions all over the place; in general we have a good handle on the paradox but there are lots of interesting ways to look at it. Zeno’s paradox is confusing because it takes these really basic notions like “Half is less than a whole” and “space and time work pretty much the way you’d think” and builds a crazy, impossible scenario with them. It’s not a hard scenario to comprehend; the real trick is resolving it.
Other times, paradoxes don’t even contain logical contradictions. Moore’s paradox, which comes to us from G.E. Moore, just says that it doesn’t make any sense for someone to say, “It is raining out, and also I do not believe it is raining out.” There’s nothing logically incoherent about someone believing that it isn’t raining when it actually is; we’re wrong about shit all the time. The problem is someone just saying that as if it makes any sense at all. It’s like, “There is a God, and I’m an atheist.” Dude, what? Pick one. Even though there’s nothing incoherent with some people being wrong about whether or not there’s a God, I already hate that guy. So how do we resolve that a perfectly good sentence with no logical contradictions at all just sounds idiotic, no matter what? Maybe we give up the idea that the meaning of a sentence is only in what the words themselves mean, which was a preeeeetty popular view not too long ago.
So sometimes, we have to get rid of some sort of premise that underlies our reasoning, a premise like “we can just name sets with predicates” or “if you add infinitely many things, even if those things keep getting smaller, it’ll go to infinity.” Sometimes a paradox doesn’t overthrow a specific premise, but a general way we go about thinking about things, like “we can always tell what an utterance means by what the individual words mean.”
The point is, Mike, there are plenty of paradoxes, and if you’re having trouble with a paradox, it means the paradox is working. There isn’t a blanket solution - they’re all doing something different. They range from “So this shit is mathematically impossible is all I’m saying” to “Well that’s fucking weird, huh?” Understanding a paradox doesn’t mean accepting it in its current form so much as understanding what might be driving the contradiction you’re trying to resolve - what does the beginning of a resolution look like? You’ll go a long way toward comprehending philosophical problems if you can learn to understand a paradox without needing it resolved right away.
If you’ve got an entire day or five to kill, Wikipedia even has a list of paradoxes.
W.V.O. Quine also gave a really good talk called The Ways of Paradox which is super accessible and highly recommended for all of you.
*In b4 Planck length - that might just be the smallest distance theoretically measurable.