mathsex: Zeno's Paradox

mothwentbad (

mothwentbad) wrote in

mathsex,
@ 2008-05-23 21:19:00

Zeno's Paradox
It's not a paradox. It doesn't make sense to me at all how anyone could think that this is a paradox. About the only thing to be learned here is that if time is discrete and space is continuous, then there cannot be continuous motion. That's hardly a paradox. Why is that supposed to be a paradox? We're just trying to move from A to B, not move from A to B while stopping to use every possible rest stop along the way.

Zeno = FAIL.

(Post a new comment)

	skreidle 2008-05-24 01:35 am UTC (link)
	It's a paradox inasmuch as any infinite limit is, which is to say, there is FAIL here, but it isn't Zeno's. Your treatment of it certainly skips over any sense of philosophy or mathematics. :P Edited at 2008-05-24 01:38 am UTC (Reply to this)

lederhosen
2008-05-24 02:10 am UTC (link)

It's not a paradox.

By definition, there are no true paradoxes, only problems that haven't been sufficiently rigorously defined :-)

It doesn't make sense to me at all how anyone could think that this is a paradox.

That's because we've absorbed a couple of thousand years' worth of philosophical process by teaching and osmosis. When you frame it in the right way it's pretty trivial, and we frame it that way by default because it's how we've seen others frame it - but without that background, the right way to approach it isn't immediately obvious.

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	en_ki 2008-05-24 01:28 pm UTC (link)
	Pls supply rigorous whatever that deparadoxifies "This sentence is false." (Reply to this)(Parent)(Thread)

	lederhosen 2008-05-24 01:58 pm UTC (link)
	http://en.wikipedia.org/wiki/Liar_paradox#Alfred_Tarski (Reply to this)(Parent)(Thread)

	en_ki 2008-05-24 02:01 pm UTC (link)
	That avoids the problem not through rigor, but through limiting the expressive power of the language. See also Gödel, below. (Reply to this)(Parent)(Thread)

lederhosen
2008-05-24 03:23 pm UTC (link)

The 'paradox' arose because we hadn't adequately defined what it means for a sentence that refers to sentences to be 'true' or 'false' in the first place. Rigorous definition inevitably runs into the fact that not all combinations of words are meaningful - but then we knew that even without the rigorous definition. Rigor just helps to point out some of the less-obvious cases.

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	en_ki 2008-05-24 03:26 pm UTC (link)
	You seem to be going in the direction of trying to fix the problem by assigning values "true", "false", and "meaningless". This also does not work. HTH. (Reply to this)(Parent)(Thread)

lederhosen
2008-05-25 12:58 am UTC (link)

You seem to be going in the direction of trying to fix the problem by assigning values "true", "false", and "meaningless".

Me? No. Godel did that about seventy years back, demonstrating that a 'meaningless' category must exist (substitute 'undecidable' if you prefer) - and if one assesses the 'liar paradox' as meaningless, then it becomes no more of a paradox than "purple monkey dishwasher".

A paradox would be if we started out with a correct premise - a man can shoot an arrow at a tortoise - and managed to prove two consequences that contradict one another (he hits the tortoise, he doesn't hit the tortoise). In this case, you start out with the implicit premise that all sentences* are either true or false - but that premise is not correct, so it's not a paradox.

*or at least, this particular one.

(Reply to this)(Parent)(Thread)

en_ki
2008-05-25 01:13 am UTC (link)

OK, I've been tiptoeing around it because this isn't remotely my specialty, but at this point I feel safe saying you need to read up a little more. You can add as many truth values as you want (meaningless, undecidable, sexweasel, orange), but it's not going to clear up the paradox.

Having added "meaningless" as a truth value, consider the sentence "this sentence is meaningless or false". If you assign it "true", it's false, so you fail. If you assign it "false" or "meaningless", it's true, so you fail.

The pain comes from the law of excluded middle, more or less. If you want to throw out that law, you can do so and still get a lot done, but first you must utter the proper shibboleth.

(Reply to this)(Parent)(Thread)

	lederhosen 2008-05-25 01:31 am UTC (link)
	So, is "purple monkey dishwasher" true or false? LOEM is only applicable when P is a meaningful statement; both your previous example and this one are not, for all that they do a better impression of it than PMD does. Note that the article you linked to specifically mentions Godel as an example of where LOEM falls down! (Reply to this)(Parent)(Thread)

en_ki
2008-05-25 01:55 am UTC (link)

"purple monkey dishwasher" is not a syntactically valid statement, so it doesn't get a truth value.

"This sentence is false" is a syntactically valid sentence (so in order to be complete, we must assign a truth value to it), but we can't assign a truth value to it in a consistent way.

The whole point of Gödel's proof is that this difference is fundamental: in any interesting language, you can't reduce semantics to syntax and still be consistent.

(Reply to this)(Parent)(Thread)

	lederhosen 2008-05-25 04:28 am UTC (link)
	"purple monkey dishwasher" is not a syntactically valid statement, so it doesn't get a truth value. Indeed - but one way to interpret the Incompleteness Theorem is to conclude that not all syntactically valid statements have T/F truth values, either. (Reply to this)(Parent)(Thread)

en_ki
2008-05-25 11:45 am UTC (link)

That's incompleteness right there. See my statement in parens.

So anyway, going back to what we're talking about, which is paradoxes: you seem to be saying that paradoxes can be eliminated. What do you think a logical paradox is, other than a syntactically valid statement that can't be assigned a truth value without violating consistency? For every set of truth values you come up with, so long as an assignment requires assigning exactly one of them, there will be an instance of the liar's paradox.

(Reply to this)(Parent)(Thread)

	lederhosen 2008-05-25 12:21 pm UTC (link)
	What do you think a logical paradox is, other than a syntactically valid statement that can't be assigned a truth value without violating consistency? I already answered this: a genuine paradox is starting from a correct premise and proving both P and not-P. Where the 'liar paradox' fails is in the premise that this statement has a truth value that is either T or F. (Reply to this)(Parent)

	langostino 2008-05-24 05:39 pm UTC (link)
	Not limiting the expressive power, but pointing out that the expressive power is much smaller than the syntactic monoid would at first blush indicate. lederhosen is right. Any syntactic system with sufficient expressive power is going to have to be capable of producing meaningless sentences, and limiting the syntax is not going to solve the problem. (Reply to this)(Parent)(Thread)

	en_ki 2008-05-25 01:15 am UTC (link)
	Well, that's not quite what Gödel said: "meaninglessness" doesn't address the issue he addressed, which was sentences that were true or false but could not be proved or disproved. I'd suggest you take it up with him, but he's paranoid and dead. (Reply to this)(Parent)(Thread)

	langostino 2008-05-25 05:12 am UTC (link)
	Uh, lederhosen might be wrong in bringing Gödel into it (oh Kurt, how abused you are!). What I'm saying is much less subtle than Gödel's theorems, because the syntactic systems I'm talking about are things like natural language, which are much more complicated than a Godelian arithmetic system. (Reply to this)(Parent)

	langostino 2008-05-24 02:16 am UTC (link)
	It's the same way that people think there must be a prime mover. We assume that everything in the infinite case works out like it does in the finite case. Frege thought the same thing, till Russell showed him up. (Reply to this)(Thread)

	gillen 2008-05-24 02:24 am UTC (link)
	But there is a Prime Mover. I've seen it. (Reply to this)(Parent)(Thread)

	issmeinhirn 2008-05-24 02:55 am UTC (link)
	bows (Reply to this)(Parent)

	subtly_modded 2008-05-24 03:17 am UTC (link)
	DO WANT. (Reply to this)(Parent)

	music_dissident 2008-05-24 03:19 am UTC (link)
	CAN HAS MOVER? (Reply to this)(Parent)

	Behold the Putzmeister! lightning_rose 2008-05-24 04:34 am UTC (link)
	(Reply to this)(Parent)

	rojonoir 2008-05-26 02:48 am UTC (link)
	Gotta say, for a thread with only a bit of math and almost no sex, this made it all worthwhile. Very sexy. (Reply to this)(Parent)

	artkouros 2008-05-24 02:21 am UTC (link)
	It is a paradox in the sense that it seems to make sense, but doesn't really. (Reply to this)

	dkogan 2008-05-24 02:35 am UTC (link)
	Yes, you are clearly much smarter than Zeno... (Reply to this)

rojonoir
2008-05-24 03:20 am UTC (link)

That reminds me of a story. A couple who shall remain nameless were fooling around. The man inserted part-way when the woman told him to stop, so he complied, remaining precisely half-way in. In this positinon they then proceeded to have a discussion about exactly at what point the loss of virginity occurs. Her hymen was lost long ago due to tampons so that was no judge. Surely halfway in didn't count. As luck would have it there was a phone on the nightstand. The woman called her friend to get a third opinion, which as it turned out wasn't especially helpful, as the amused friend was unable to give a rigorous definition of virginity or lack thereof.

Ah, but if only she had called Zeno. She'd be a virgin to this day.

(Reply to this)(Thread)

	langostino 2008-05-24 01:40 pm UTC (link)
	Her hymen was lost long ago due to tampons Uh, what? (Reply to this)(Parent)(Thread)

	rojonoir 2008-05-24 03:31 pm UTC (link)
	http://www.beinggirl.com/en_US/questiondetail.jsp?ContentId=ASK173 (Reply to this)(Parent)(Thread)

	langostino 2008-05-24 05:32 pm UTC (link)
	Still, "an active childhood" seems like a more plausible reason to give. (Reply to this)(Parent)(Thread)

	vvvexation 2008-05-27 02:56 am UTC (link)
	What do you mean "plausible"? Who's supposed to be convincing whom of what? (Reply to this)(Parent)(Thread)

	langostino 2008-05-27 03:08 am UTC (link)
	Isn't the point of fiction to convince? (Reply to this)(Parent)(Thread)

	vvvexation 2008-05-27 06:04 am UTC (link)
	No, generally that's exactly what it isn't. Now, if you're implying rojonoir's story is made up, that's a different matter. But would it make a difference if I told you I know someone who that same story also happened to (minus the tampon part)? (Reply to this)(Parent)(Thread)

	langostino 2008-05-27 12:03 pm UTC (link)
	In that I wouldn't go "bwa?" when it came to the tampon part, yeah. (Reply to this)(Parent)

	mothwentbad 2008-05-25 12:37 am UTC (link)
	This doesn't sound like the best sex ever had to me. (Reply to this)(Parent)

	czarandy 2008-05-24 04:04 am UTC (link)
	You don't really address the paradox at all. Time being discrete has nothing to do with it. (Reply to this)(Thread)

	mothwentbad 2008-05-25 12:36 am UTC (link)
	I don't address the paradox because there isn't one. If you believe in the real numbers and you believe in multiplication by three, then you can describe where something moving at 3 feet per second is at time t. There exist bounded strictly increasing subsequences. Paradox! (Reply to this)(Parent)

	kvschwartz 2008-05-24 04:23 am UTC (link)
	Considering it took nearly twenty-five CENTURIES for mathematicians to work out all the knots in Zeno's paradoxes (and even NOW philosophers are STILL debating them) ... I would say Zeno does not "fail." (Reply to this)(Thread)

	langostino 2008-05-24 01:41 pm UTC (link)
	I donno, I mean, philosophers are kind of notorious for kicking little nuggets of fail around. (Reply to this)(Parent)(Thread)

	mothwentbad 2008-05-25 03:19 am UTC (link)
	Yeah, philosophers are subject to Sturgeon's Law, ~~even~~ especially the famous ones. (Reply to this)(Parent)

	mothrinventor 2008-05-24 03:05 pm UTC (link)
	doesn't this basic problem still have great relevance to quantum mechanics and modern physics? (Reply to this)(Thread)

	mothrinventor 2008-05-24 03:06 pm UTC (link)
	...and string theory? (Reply to this)(Parent)

	langostino 2008-05-24 05:34 pm UTC (link)
	Uh, no, because the whole idea of quantum anything is that everything in sight is discrete. Discretizing space is actually the easiest way out of Zeno's paradox, because if you're in Z³, eventually there is no "halfway" anymore. (Reply to this)(Parent)

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