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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,

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

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 (1/2 + 1/4 + 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

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

*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 (1/2 + 1/4 + 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

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.----

Wikipedia has articles on Russell's Paradox, Zeno's Paradoxes, and Moore's Paradox, all of which have examples and more fleshed-out resolutions of the individual paradoxes. 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.

Bro!, Russell's paradox was a response to Frege's "definition" of what a natural number was (which was his life work). Set theory didn't solve this problem (neither did Russell or Frege ... or anyone still today).

ReplyDeleteRussell's paradox did a whole bunch of things, including fuck up Frege's program, but it also was trouble for Cantor's naive set theory. It lead to the development of the ZFC axioms, among other things, so we've definitely solved what I'd say is the

Deletemain problemof Russell's Paradox.Russell told Frege about the paradox more than a year after he first formulated it, though; Frege was trying to give an ontology of numbers which ended up not working. But people certainly have ontologies of numbers that aren't hurt by Russell's Paradox.

Bro, there is no such thing as *Cantor's* naive set theory. Cantor never accepted or proposed the so-called "naive comprehension principle" (NCP) that is characteristic of naive set theory that says that, for any description, there is a set of things satisfying that description. Almost two decades before Russell discovered the paradox in Frege's system (which, in effect *did* embrace the NCP), Cantor was already distinguishing between ordinary sets that can be assigned an infinite cardinality and overly-large, "absolutely infinite" collections (like the collection of all ordinal numbers) that cannot and hence cannot be assumed to behave like ordinary sets. In doing so, he was anticipating the modern distinction between sets and so-called "proper classes" that is now a standard part of modern ZFC-style set theories and which is typically appealed to in the resolution of the set theoretic paradoxes. Naive set theory (as characterized by the NCP) and Cantorian set theory are two very different things. An excellent source for all of this is Michael Hallett's _Cantorian Set Theory and Limitation of Size_.

DeleteSpot on professor PB! Forgot about the ZFC definition(s).

ReplyDeleteI think (I think...) that Godel responded to this by saying "Fuck you, I ain't givin' you a definition for something you already understand...in German. (How analytic of him). To which Russell was like... "Shit! He has a point." (not ... in German). Maybe that's what I was thinking of when I wrote today? Or maybe O'm just flat wrong again...

Hey, thanks a lot man. I kinda get what paradoxes mean now (or what they don't mean? lol) The stuff on Moore's paradox is pretty interesting and the last part made a great summary. I gotta do some of my own research on these now. Haha either way, thanks bro, really cleared up a whole bunch of questions.

ReplyDeleteMay you never stop philosophizing.

Obviously, out brains evolved to deal with Xeno's paradox. My theory is that whenever we decide to go somewhere, we unconsciously adjust our destination to be twice as far away. Then, when we get halfway there, we stop.

ReplyDelete