well then we just might have to jump the final quark distance. Anyway, whether it's possible or not, my explanation worked either way Please Register or Log in to view the hidden image!
This is my first post, so I just want to say, "Hi". You sure have been discussing Zeno for a long time here. I have read arguments from number theory and quantization of space, but I don't think either of those are necessary, particularly the latter. Zeno's argument fails in continuous space as well, as you will see in the following review. Zeno sez: 1. Any object in motion between two points must first pass through the midpoint. 2. Any such distance can be so divided, ad infinitum. 3. From (2), it follows that there are an infinite number of lengths through which the object must pass. 4. It takes a finite amout of time to pass through each length. 5. By (3) and (4), it follows that it takes an infinite amount of time to traverse any distance. Conclusion: Therefore, motion is an impossible illusion. Zeno's mistake is in step 5. This deduction involves the evaluation of an infinite series that he assumes will diverge, when in fact it does not. Here's the right way to do it. The distance between the two points (call it "L") can be divided thusly: L/2+L/4+L/8+L/16... If the object is not accelerating, then we can say that "v" is the constant velocity that the object appears to have, and T is the total time in which it appears to traverse the distance. Thus, the total time to traverse each subinterval is: T=L/(2v)+L/(4v)+L/(8v)+L/(16v)... T=(L/v)*sum(n=1 to infinity)(1/2)^n Sorry, I don't know how to write mathematical symbols here. Anyway, the above series is a geometric series that converges to 1, giving us T=L/v, as expected. Like I said, Zeno's argument depends on this series diverging to infinity, and it does not. Thanks for reading, Tom
Hm, I see this useless thread still lives on. Zeno's life was a complete and utter waste of time. He forgot one teeeeeeeny minor thing in his theories...the arrow just fucking moves through the room!! His babbling about never reaching the end of the room is destroyed by simple empirical evidence. Pure reason without empirical/realistic backing is pure trash.
Zero, <i>Zeno's life was a complete and utter waste of time.</i> Let's see whether people still remember <b>you</b> a couple of thousand years from now.
I have yet to make my earth shattering mark on this world. Please Register or Log in to view the hidden image! It shall come. When it does, everyone will feel the impact.
Thank you! You just proved space and time are quantized, like I said, and confirmed my proof that Zeno's paradox is flawed.
No, I did not prove the quantization of space and time at all. As to space, the I assumed that space was continuous in that it is infinitely divisible. As to time, all I said is that, for each interval, of the distance travelled, the time is the ratio of the interval length to the constant speed, so time is again continuous in my analysis. As an analogy, consider the range of potential energies in the Sun's gravitational field, as described by Newton. I can sub-divide that range just as I sub-divided the above length. Does that mean the energy levels are quantized in Newton's theory? I think it is pretty clear that they are not.
But you did. How can it take a finite amount of time to travel an infinite distance or vice versa? Obviously it shows that you can't subdivide it infinitely, therefore it's quantized.
You know what I mean! You said it takes a finite amount of time to traverse any distance. And you can't subdivide it infinitely. If you add up an infinite number of infinitely small distances, how much distance do you have? None? Infinite? Either way, it can't take a finite time to travel it.
Well, you can't do anything infinitely, so this is true by default. However, the requirement of quantized space is that it cannot be subdivided without limit. Is this the case here? Look at the summand in the infinite series in Zeno's paradox: (1/2)^n, which can be arbitrarily small. In other words, there is no "smallest distance". If you subdivide the space to 10,000 intervals, I can always divide it again a 10,001st time, and so on. Thus, the space in this model is continuous.
But you can't! You can't always divide one more. Eventually you get to the point where you can imagine dividing one more, but the distance has no meaning.
problem is that that distance is different for every person Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
I'm not disputing the truth of this. I'm telling you that I did not PROVE it, neither can it be inferred from what I said here. My analysis of Zeno's paradox proves only one thing: if you assume a continuous 1D space with motion at constant velocity, there are no absurdities, contrary to Zeno's conclusion. You are IMPOSING your quantum space conclusion on the analysis, as obviated by your assertion above.
Sorry to drag an old thread back. I was following this thread but am stuck on a couple of things: 1) Merlijn, in your example, you added that the distance between B and C is zero. That would mean B=C which means they would be the same point. But your illustration clearly shows a small gap - a distance. This part I did not get. 2) Is there a consensus yet as to whether the following is true: 0.999... = 1 0.333... = 1/3
Depends on the mathematics you are using! If you are using standard analysis then 0.99999.. equals exactly 1. If you are using non-standard analysis by Robinson 0.99999. differs from one by an infinitesimal amount. Greetings, Han.
I think I have the answer Xenu: If the steps are in the ratio of 1/2, then the sum from 1 to infinity=1. since we cannot take infinite steps, he would never reach our detination. if we changed the ratio to 1/(1.9), the sum would be [1/(a-r)]-1 =1/(1-(10/19))-1= 10/9=1.111111111111....... thus, by using a ratio greater than 1/2 we reach our destination (1) in a finite number of steps (although still a very large number) I think this is what Xenu wanted to know. edit: oops! bad example i used. in my example we would actually hit the wall ont the 4th step. if we used a number slightly greater than 1/2 for the ratio, such as 1/1.9999999, it would take more steps. I'm too tired to figure it out, but I'm sure a math wiz could derive a formula to express the number of steps based on the relationship of the ratio to 1/2 some figure it out please so I dont have to.... I gots other math I should be doing I havnt even started....If not I guess I'll have to later.
Han Baumer: Sorry, but after reading this ( http://www.somethingawful.com/articles.php?a=88 ) article, (and I hope thats not you) , the "greetings at the end of your post is too funny. Usually you' start with greetings, and well, you'll see when you get to the second page. Greetings
If Zeno's reasoning was right , than he couldn't even get halfway the room for the same reasons, so obviously Zeno was wrong.
