Just wondering what all solutions you guys come up with these . I think it's incredible that people have been debating these for more than two millenia... and still people cannot agree on the correct answer, even though we all know Zeno was wrong. The most famous one is called "Achilles." Here's the set-up: Achilles is the fastest man in the world, and runs at 10 meters per second. He's going to race a tortoise, and to give the animal a chance, he gives it a 100 meter head start. After 10 seconds, Achilles will be at the tortoise's original position. But in that time, the tortoise will have travelled another 10 meters. After another second, Achilles will have travelled the 10 meters, but the tortoise will still be a meter ahead.... and on and on. Thus, Zeno said, Achilles could never catch the tortoise. How would you fellows explain it? Probably the most interesting day we've had in my metaphysics class!
I don't get it... Unless the race is a 110 meter race or there is a time limit of 11 seconds, Achilles will overtake the tortoise in 12 seconds.... Here's a better paradox: There's a sign in a barber shop that says: "I shave all those and only those who don't shave themselves." So, who shaves the barber? x34
10*t=1*t +100 achiles=10t tortoise=t+100 t=0 achilles =0, tortoise =100 t=10 achiles=100 tortoise= 110 totoise-achilles=10 t=10+1=11 achilles=110 tortoise= 111 t=100/9 achilles=1000/9 tortoise=100/9+100=100/9+900/9=1000/9 So after 11 and 1/9 second, achilles catches the tortoise.
Joe Joe: That's Bertrand Russell's answer. But it begs the question. The problem isn't that Achilles will overtake the tortoise. We all know it will. The problem is how Achilles performs an infinite series of actions in a finite time. Russell attempted to solve the question by proposing that there was an "actual infinite" at which the series converaged to a point. Besides the obvious problem that this is an approximate, and not precise, it doesn't explain how Achilles reaches it in the first place. Incidentally, X34's paradox is a real one. That's an example of it, but the real problem is: Can there be a set of those entities that are not in a set?
it doesn't explain how Achilles reaches it in the first place. WELL DUH. How he reaches it? YOU KNOW. Huh. Right? On foot.
I think Zeno's only intelligence was in getting people to waste their time debating this. If you try to debate it within the confines of his logic, it doesn't make sense. We KNOW achilles will overtake the tortoise even though Zeno's logic dictates that he won't. Look at it with some good 'ol common horse sense, and it all becomes clear. If the turtle is moving slower than Achilles, then Achilles WILL overtake and pass him. The theory that all Achilles is doing is spending his time trying to the point where the tortoise was last (while the tortoise is continually moving ahead) is bull****. Achilles stride will take him to the point where the tortoise was last and with one step, he'll overtake and surpass him. It's easy to see the truth if you can throw out the BS.
haven: When I had Calculus I 10 years ago, we talked about Zeno's paradox in defining the concept of a limit. Here's the one that I remember: Say that you are in a room and are 100 feet from the door. Let's say that every step you take is half the distance to the door. So, your first step consists of travelling 50 feet. Then your next step will consist of travelling 25 feet. The next is 12.5 feet, and then 6.25, and so on. Eventually, you get right next to the door, but never to the point of being able to get past it (theoretically). I always thought that was cool to think of it that way. It is better to actually show this then tell it. I think the answer to x34's paradox is the barber himself. The only reason why I think I know that is because I have heard of this one before although it seemed to have to do with shaving people in a village.
Well obviously Zeno's stupid thing just implies that time converges essentially to a stopping point which it doesn't. Performing an infinite number of operations in a finite time is how the entire world works. I move my arm an infinite number of infinitely small distances every time I shoot a basketball. Zeno's paradox requires time to slow down more and more as it reach 11 1/9 seconds (which never allows Achilles to get to 11 1/9 seconds to pass the tortoise) and that doesn't happen and that's why it's seems to me to be such an incredibly dumb problem to debate. Now the barber one is good. I think it's the barber cuz he thinks he doesn't shave himself so he shaves a little off, then realizes "oh damn, now I can't shave me" but then he realizes he would look dumb with just a little shaved off so he says screw the rule and shaves himself anyway.
Heh... but the question isn't if Zeno was wrong. But essentially whether time and space are discrete or not. Same thing with francis for perez's response. You can perform infinite actions in finite times? How? I like one of two answers: 1. mathematics is fallible as a symbolic representation of reality 2. Aristotle's answer: there's a difference between potential and actual divisions of space. I thought people would be more interested in this than they were. Sorry . The barber question is unsolvable if worded correctly. Russell's paradox has no true resolution .
The mathematical solution is what's called an infinite sum... And whoever said to take the limit of the sum is correct. Bonehead Zeno just never believed in the sum of infinite numbers yielding a finite answer. That's his problem... the rest of the world has moved on.
Oh, and a co-worker says "Watch the movie I.Q. with Meg Ryan and Tim Robbins -- it's got an example of Zeno's paradox in it".
DoD: You're betraying your age . Whitehead and Russell believed that calculus could solve the paradox... but Wesley Salmon revealed that it really didn't! The problem is that Zeno's asking an illegitimate question. Trying to resolve it within his framework is impossible... and a waste of time!
Another way of disputing the merits in the argument is to point out that there is an assumption ( in this case time/space ) which in itself prevents resolution, but which has no foundation in reality..The best example is of the one mentioned about continually halving the distance...When it says " assume that you walk half the distance to the door with every step " it has already broken from reality, and made resolution impossible...It might just as well be argued " Assume that there is no solution to this problem, how do you solve it?" Zeno's premise, while more elegantly argued, contains the same assumption which precludes resolution... haven....keep up the interesting posts, though..
"I shave those and only those who don't shave themselves". If the barber shaves himself, then that means he's shaving someone that shaves himself, so you'd be wrong.
JAG: Do you object to thought experiments in general, then? Or just ones that don't are logically self-contradictory? I sort of like them myself...
uh...I wasn't being sarcastic, I really meant 'keep them coming"...So, no, I don't object to thought experiments...and ones that are logically self-contradictory, if addressed from a logical standpoint, are merely non-starters, not objectional...
You described what I was trying to explain as what I knew as Zeno's paradox. You also did a better job of explaining it. DoD: I know that I have heard that about the barber but it was shaving people who lived in a village. The way x34 has worded it makes me agree with haven that this is unsolvable.
I haven't read the rest of the posts, but this one is easy: "After another second, Achilles will have travelled the 10 meters, but the tortoise will still be a meter ahead . . . " The tortoise will always be 1 meter ahead. It doesn't say that Achilles will travel 10 meters and the tortoise will travel 1 . . . it says that every second, Achilles travels 10 meters, but the tortoise will always be 1 meter ahead . . . always . . . 1 meter ahead . . . you could go forever, and the tortoise will always be 1 meter ahead.
