therefore, Achilles travels 10 meters, and the tortoise travels 1 meter more, each time . . . thus, 11 meters I think this is correct . . .
Hmm . . . well, I just read the posts . . . It seems so simple . . . but I must be missing something?
No. It says that the tortoise moves at a rate of 1 meter per second. haven- I'm sorry if I'm not getting this. Can you please explain it to me again? After 10 seconds elapse, Achilles would be 11 meters behind the tortoise. If one more seconds elapses, the tortoise would be 11 meters from its starting point, and achilles would be 10 meters from the tortoise's starting point. (1 meter behind the tortoise). So if one more second elapses, Achilles will be 8 meters ahead of the tortoise. Is that not correct?
One more thing. I really don't see any application of limits here. The rate of distance traveled by Achilles and the tortoise is constant, right?
What are these infinite series of actions? Running for 11 seconds and change? Damn. If thats infinite, than I can run for like 6 infinite series of actions.
That's not what the original question says though. It doesn't say that the tortoise moves at 1 meter per second, it says that: "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. <b>After another second, Achilles will have travelled the 10 meters, but the tortoise will still be a meter ahead.... and on and on.</b> Thus, Zeno said, Achilles could never catch the tortoise. " It never says that the tortoise goes at a constant speed . . . The problem states that: Achilles runs at 10 meters per second. The turtle starts out 100 meters in front of Achilles After 10 seconds, Achilles had gone 100 meters, and the tortoise had gone 10. 1 second passes, Achilles goes 10 meters, the turtle remains 1 meter ahead and so on and so on the <b>so on and so on</b> is what's the key. It's kindof like programming. It loops the previous statement -- "1 second passes, Achilles goes 10 meters, the turtle remains 1 meter ahead" Every second that passes, Achilles will have gone 10 meters, but the turtle will remain 1 meter ahead. Thus, the turtle's speed must have changed. No where does it say that the turtle remains at a constant speed, only that after 11 seconds, it's 1 meter ahead of Achilles. program snippet as it would be seen in C++: ---------------------------------- const int Achillesspeed = 10; Achillesdistance = 0; Turtledistance = 100; for (int x = 1; x >= 10; x++) { Achillesdistance+=Achillesspeed; Turtledistance++; } while (!finished) { Achillesdistance += Achillesspeed; Turtledistance = Achillesdistance+1; } ----------------------------------- Achilles' speed is a constant, but the turtle's is not. If this is the correct wording of the problem, then the above program would represent what the problem is trying to show you. Nowhere does it say that the turtle remains at a constant speed. Thus, he went at 1 mps (Meter per second) for 11 seconds, then at 11 mps after that, because he always remains one meter ahead of Achilles.
"Another version of Zeno's paradox involves a race between Achilles and a Tortoise. Achilles can run 10 times as fast as the tortoise and therefore gives the tortoise a ten meter head start. However, if the tortoise has a ten meter head start how can Achilles ever catch him? By the time Achilles reaches the 10 meter mark, the tortoise will be at 11 meters. By the time Achilles gets there the tortoise will be at 11.1 meters and so on. This process of looking at where the tortoise will be when Achilles catches up to where he WAS can be repeated indefinitely creating an infinite sequence of snapshots all showing the tortoise still ahead. Therefore, Achilles, even though he runs ten times as fast as the tortoise, will never catch up. " That is the exact wording. Your interpretation is incorrect. But its still dumb. you can look at it 2 ways. 1: By the time Achilles reaches 11.1, the tortoise will be at 11.11. or 2: By the time the tortoise reaches 11.2, Achilles will be at 12 meters.
I've done some looking around, and the original problem is not necessarily the best wording . . . here are some other links on this subject: http://www.intercom.net/user/kathleen/zeno.html http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp http://www.mindspring.com/~mfpatton/zeno2a.htm It makes sense -- basically, if you take a snapshot in time, every time that Achilles will have reached the tortoise's previous location, the tortoise will have moved an extremely small amount further than where he had been -- thus, Achilles can never truly catch up. It's all about taking snapshots in time, and the snapshot and the distance becoming integrally smaller and smaller. The links explain this. Very cool
Actually, thinking on option 1, it says when Achilles gets to 11, the tortoise will be at 11.1. Fine. But when the Achilles gets to 11.1, the tortoise will get to 11.11. ARG. This is SOOOOOOOOO not logical. Logic states that because Achilles moves faster, he will eventually overtake the tortoise which moves slower.
It's not cool. Its stupid. Snapshots of time? What the **** is that. No. I don't buy Zeno's bull****. Achilles goes faster, he will overtake the tortoise. I will take your camera that is taking these "snapshots", and dump a huge cleveland steamer on the lens.
How is my interpretation dumb? I'll grant you that it's not correct, but based on the wording of the original problem posted here, it never states that the turtle will be going at a constant speed. It says that Achilles does, but never states that the turtle does. Had it said that, obviously my point would have not been correct. Now, if we look at it from a different standpoint (as I've seen through looking at other websites on this issue), it's basically a problem of diminishing values, but it does make sense, and it's a cool paradox. In our real world, we would see that the turtle would be passed up, but let's look at it this way: Achilles = 10 mps Turtle = 1 mps Turtle = 100 meters Achilles = 0 meters After 10 seconds: Achilles = 100 meters Turtle = 110 meters 1 second later: Achilles = 110 meters Turtle = 111 meters 0.1 second later: Achilles = 111 meters Turtle = 111.1 meters 0.01 seconds later: Achilles = 111.1 meters Turtle = 111.11 meters 0.001 seconds later: Achilles = 111.11 meters Turtle = 111.111 meters and so on, and so forth -- By the time that Achilles has caught up to where the turtle is, the turtle has moved a fraction further.
Lessons: "<i>Part Two: The Lessons Lesson 1--Zeno is very annoying. But he knows this. He is advocating the somewhat radical thesis that we ought not to follow our senses, but reason and logic in formulating beliefs about the world. The senses and logic conflict, the senses are the less reliable of the two, so they and their testimony are disregarded. His arguments get most of their power from appeal to the foundational assumption of Logic--The Law of Noncontradiction. This law is the thesis that a claim is true or is false. Nothing can be true and false at the same time. This law is exploited in reductio ad absurdum arguments. The strategy of the reductio is to accept a premise for the sake of argument and derive a contradiction from it. Once this is accomplished, the Law of Noncontradiction tells us the original assumption was false. Parmenides and Zeno thought something like the Law of Noncontradiction applied to objects or beings as well. They believed that no statement of the form "_____ is not" or "_____ does not exist" made sense. For a term to occupy the subject position of a sentence, it must be a term which refers to a thing--an existing thing. Their quick answers: "What exists? Everything." "Are there things that don't exist? Of course not." This is more than mere grammar. I urge you to think about these issues when you struggle with the paradoxes. Lesson 2--It doesn't have to go this way. But the alternative isn't very promising. We don't have much left if we merely rely on the senses. Science stops in the age of alchemy, for example. Almost all of our intellectual endeavors involve repudiating these senses to some degree or another, but why stop halfway. If you want to refuse some of Zeno's conclusions, well, fine, so do I. But I think to be respectable you have to answer Zeno on his own terms (with arguments) and explain why you don't go all the way with him when you do favor reason over the senses in some cases. This is harder than you might think. Lesson 3--You can't just show Zeno the motion of your fist zooming towards his face, or the plurality of his teeth scattering on the floor, no matter how satisfying this would be. First of all, he's dead. But more importantly, it doesn't matter to him and it shouldn't. He doesn't deny that things look like they move--he wouldn't be arguing if all was as it appeared to be. He is arguing rather forcefully that appearances should be denied. Showing him more appearances won't help--he thinks they are irrelevant. If all you have on your side is more appearances, you've lost the fight. There are plenty of other lessons to be taken from these issues and the others we have been covering in class of late. You should think about them for their own sake and because they are issues you should address in your papers.</i>" from: http://www.mindspring.com/~mfpatton/zeno2a.htm
I didn't say your interpretation was dumb. Zeno was dumb. I understand limits. Even if the turtle's speed is variable, Achilles would still overtake him. He is moving faster.
It only works with 1s your way. What about this? Achilles = 10 mps Turtle = 1 mps After 10 seconds: Turtle = 100 meters Achilles = 0 meters 1 second later: Achilles = 110 meters Turtle = 111 meters 1.2 seconds later: Achilles = 112 meters Turtle = 111.2 meters Turtle has been passed. Bada bing, bada boom.
True, there is another way to look at this, but you can't determine the instant at which Achilles' location equals the turtle's. In order to pass the turtle, he has to equal the turtle first. Yet that instant is what diminishes to infinity.
You can predict when Achilles will have passed the tortoise mathematically, but not exactly when or more importantly how. You're proving something we already know... namely, that Zeno was wrong. But that doesn't explain what's going on. Sorry for the confusion vengeance... I thought that once I mentioned the regression from 10 to 1 sec intervals, you guys would see that. Should have made myself more clear. Nomar: of course Zeno was wrong. That's why we read Aristotle instead of Zeno. In fact, we don't even possess the original writings of zeno. We possess writings of Aristotle in which he describes Zeno's paradoxes. He attempts to rebut them. There have been many attempts to explain the paradoxes. Here's another one (it goes further than Achilles): This one's called the regressive dichotomy. Achilles is running to a finish line. But before he can get there, he must get to the half way point. Before he gets to the half way point, he must get to the 1/4 point. Before he gets to the 1/8 point, he must get to the 1/16th point. And so on and so on (there, vengeance )... Hence, Achilles can never reach his destination... indeed, he can never actually move, because he has to perform an infinite series of actions to do anything. But wait, you see! Movements are discrete, and space and time are not infinitely divisible? Then here's another paradox for you, to counter that view: An arrow is in flight. If space/time is discrete, how does it ever leave it's current position? In a single instant, it must be in a single place. Otherwise, the instant would not be discrete and time would be continual, which runs counter to the former paradox. So when does it move? Between instants? But time, and therefore movement, can't exist between instants. There's also no velocity. How could there be, if the arrow is perfectly still in any given instant? Zeno was a Monist, probably, in the tradition of Parminedes, who wanted to prove that everything was One. Time, change... didn't exist. I can remember two others off the top of my head, but that's all I'll leave you with for now. Remember: we *know* he's wrong... and I'm not arguing his right. The question is, why he's wrong, and it's still an unresolved question today. There are different answers that many support, but not everyone agrees on any one solution. I've enjoyed hearing the different perspectives.
Actually, if you want to read the problem that way, you made a slight mistake. For the turtle to remain 1 meter ahead of Achilles, he must first travel at 1 m/s for the first 11 seconds. Then he instantly changes speed (not accelerates) to 10 m/s. He will be going at the same speed as Achilles, but since after 11 seconds he was 111 meters from the start, and Achilles was at 110 meters, Achilles would remain 1 second behind him based on those assumptions. That being said, the problem is, as others have said, bullcrappy. The only difficulty is in measuring the exact moment in time where Achilles passes the turtle, since it is an infinitesimally small fraction...11 s + 1/10 s + 1/100 s + .... Zeno's math skills may not have been able to handle calculating this, but we all know where the turtle is after 12 seconds and where Achilles is after 12 seconds. That's why it's ridiculous to even debate it. I've read some of the Greek's writing, e.g. Plato and it seems that too much of it is just so much mental masturbation, if you will. Twisting words to make the nonsensical seem to make sense is just a waste of time. It's non-productive. But if you want an example of a paradox from real life, here's one from my job - this is what I have to deal with: Boss: Given that you have no parameters for the project, how long will it take to develop it? Programmer: ....(@#%^@#$^#$@#).... two weeks? Boss: Ok, I'll hold you to that Programmer: @#$^!@#$^@#$@#$^@#$^@#$!!!
Ah... you're still begging the question . Then again, if you don't like thought experiments in philosophy, you're not going to like stuff like this. Incidentally... Plato's work formed the philosophical framework that underpins Western civilization. I think it's very difficult to dismiss it as you do... I tend to enjoy Aristotle much more than Plato, because he's so much more sophisticated. But you can't have Aristotle without having Plato first. And without Aristotle... well, history would be radically different.
