|Physics: The Zeno Paradox|
|Released by matroid on Tue. March 21, 2006 17:03:40 [Statistics]|
Written by Nodorsk - 30999 x read [Outline] Printer-friendly version - Choose language
\ (\ begingroup \)
About 2500 years ago, Zeno of Elea set up a paradox that will be familiar to many. If the fast-footed Achilles gives the turtle a head start in a race, he will never be able to overtake them. With this and a few other examples, Zeno argued that the knowledge we perceive with the sense organs need not necessarily be true. The paradox with Achilles and the turtle should prove that movement is impossible, as there are no contradictions or logical errors in the presentation of the situation. How can the paradox be resolved?
The paradox Achilles wants to race a turtle. Since the turtle is slower, it gets a head start. Now the race starts. Achilles reaches the point where the turtle is at the start time. In the meantime, however, she has made some headway. Achilles has now reached this point and the turtle is a little further ahead. So the race continues and Achilles will not be able to catch up with his opponent, because when Achilles reaches the last point of the animal, he is already a bit forward.
Solution of a student Of course, the task can be solved with the simplest school mathematics if we ignore the individual time intervals. Achilles with the speed v_a gives the turtle a lead W with v_s. If Achilles overtakes the animal after the time T, he has covered a distance S, therefore T = S / (v_a) = (S - W) / (v_s) S - W = v_s / v_a S => S = W 1 / (1 - v_s / v_a) = (W v_a) / (v_a - v_s) Achilles has the turtle after the distance (W v_a) / (v_a - v_s) caught up and after the time T = W / (v_a - v_s). But that doesn't solve the paradox at all, we just assumed that he would catch up with them and that the time was finite. But is that also the case?
Solution with the help of a number Zeno concludes that Achilles never overtakes the turtle. A mathematician these days would write T \ textrightarrow \ inf. Zeno comes to this statement because there are infinitely many time intervals before overtaking occurs, so we can write T as T = sum (t_k, k = 1, \ inf). Now we want to calculate T, for this we need the individual t_k and finally want to see whether the series diverges or not ... Achilles reaches the starting point of the turtle after t_1 = W / v_a, while the turtle has covered the way t_1 v_s = W / v_a v_s. Achilles bridged this distance after the time t_2 = W / v_a v_s 1 / v_a, while the turtle is now further, namely by t_2 v_s = W / v_a v_s ^ 2 1 / v_a. Hence we get t_3 = W / v_a v_s ^ 2 1 / (v_a ^ 2). Analogously, we get the remaining t_k, so that we can write T as T = t_1 + t_2 + t_3 + .... T = W / v_a + W / v_a v_s 1 / v_a + W / v_a v_s ^ 2 1 / ( v_a ^ 2) + ... T = W / v_a (1 + v_s / v_a + (v_s / v_a) ^ 2 + ...) T = W / v_a sum ((v_s / v_a) ^ k, k = 0 , \ inf). Now we want to know whether T \ textrightarrow \ inf converges or not ... Of course, some readers will now know that this is the geometric series and that we could actually get the limit immediately if we entered the values into a formula . But we don't want to do this, we want to see how you get it. Perhaps Zeno could have calculated it too ... Let us put q: = v_s / v_a, da v_s < v_a="" ,v_s=""><> 0 and v_a <> 0 is q \ el] 0.1 [. Thus T = W / v_a sum (q ^ k, k = 0, \ inf) = W / v_a lim (n -> \ inf, sum (q ^ k, k = 0, n)). Now let S_n: = sum (q ^ k, k = 0, n). From this it follows S_n = 1 + q + q ^ 2 + q ^ 3 + ..... + q ^ n S_n = 1 + q (1 + q + q ^ 2 + ... + q ^ (n-1) ) S_n = 1 + q (S_n - q ^ n) S_n - q S_n + q ^ (n + 1) = 1 S_n = (1 - q ^ (n + 1)) / (1 - q). Since q \ el] 0,1 [, we get lim (n -> \ inf, q ^ (n + 1)) = 0. I don't want to give the proof for lim (n -> \ inf, q ^ (n + 1)) = 0 here. However, one can already assume it intuitively that q> q ^ 2> q ^ 3> ... and thus lim (n -> \ inf, S) = lim (n -> \ inf, (1 - q ^ ( n + 1)) / (1 - q)) = 1 / (1 - q). Strictly speaking, this even applies to all q \ el] -1,1 [, but since this is not necessary for our case, it is sufficient to know that it applies to q \ el] 0,1 [. Now let's see what this calculation gives us and insert: T = W / v_a sum ((v_s / v_a) ^ k, k = 0, \ inf) T = W / v_a 1 / (1 - v_s / v_a) = W / v_a v_a / (V_a - v_s) T = W / (V_a - v_s). It is of course - as was to be expected - the same result as above.
Final remark As we have seen, Achilles actually overtakes the turtle. With this paradox, which was indissoluble at the time, Zeno wanted to show that the world is not discreet by assuming that it is and that a contradiction arises with the help of the race between the two opponents. With further paradoxes he showed that the world is not continuous. The infinite that penetrated the analysis of motion in this way worried mathematicians and physicists up until the 17th century, thereby leading to differential and integral calculus, strongly associated with the names Leibniz and Newton, who developed the art, who Determine the slope of the tangent to a curve or the area of a body that is bounded by a curve. Since then we have been able to describe the movement of macroscopic objects very well. But, to anticipate it, we still don't know whether the world is discrete or continuous. The emergence of the theory of relativity and quantum physics raises new problems as well as knowledge with regard to this question. Ludwig Wittgenstein, Austrian philosopher of the 20th century, noticed a "mistake" in the argumentation of the paradox and other mathematical constructions. He claimed that terms like limit value or series are inherently consistent, but have nothing to do with reality. Just for fun, we can pick up a runner and a turtle and let them both race. If the time we calculated coincides with the measured time, we can assume that we are correct. But is that actually proof? To what extent does mathematics explain nature and, in general, is mathematics a discovery or an invention of man? There are still many unanswered questions that are at least worth considering. The ancient Greeks are in no way inferior to us in their curiosity and interest in understanding nature and many questions from then are still very topical today. I would like to end my article with a quote about Zeno from Plato: "Zeno spoke with an art that made the same things appear similar and dissimilar to the audience at the same time, one and many, immobile and mobile."
\ (\ endgroup \)
|Get link to this article Printer-friendly version - Choose languageComments|
There is no pdf file for this article
This article is registered in the directory of the Alexandria working group:[The Alexandria Working Group catalogs the articles on the math planet]
|"Physics: Zeno's Paradox" | 31 Comments|
|The authors of the comments are responsible for the content.|
Re: The Zeno Paradoxby: AimpliesB on: Tue. March 21, 2006 20:36:25
|\ (\ begingroup \) Kudos. \ (\ endgroup \)|
Re: The Zeno Paradoxby: FlorianM on: Tuesday March 21, 2006 20:45:31
|\ (\ begingroup \) Hi Nodorsk, wonderful first article! Keep it up! Greetings Florian \ (\ endgroup \)|
Re: The Zeno Paradoxfrom: Ex_Mitglied_40174 on: Tuesday March 21, 2006 20:50:45
|\ (\ begingroup \) I find the paradox of the rows in motion much more interesting (google is happy to help those who do not know it). It shows that the problem of the relativity of movements in different frames of reference already preoccupied the ancient Greeks. Greetings your Tarbaig, as always too lazy to log in \ (\ endgroup \)|
Re: The Zeno Paradoxby: Spock on: Tuesday March 21, 2006 21:53:55
|\ (\ begingroup \) Hello Dorsk, this is a successful, beautiful first article from you. I like it because it raises at least as many questions as it answers, and because of the picture below your "series solution": It really is one greek tortoise? Greetings Juergen \ (\ endgroup \)|
Re: The Zeno Paradoxby: Wauzi on: Tuesday, March 21, 2006 11:15:01 pm
|\ (\ begingroup \) Hello, as mentioned at one point in the article, this paradox is based on a deeper problem. Primarily the question of continuous or discreet. But thought further, this is a consequence of the concept of infinity, which was not even rudimentarily understood at the time. Neither on a large nor on a small scale. We find this in many places in contemporary Greek science. Most well-known is Democritus' concept of the atom, which of course was not based on scientific considerations, but was also a consequence of the notional acceptance of infinity (here on a small scale). One must go a long way in time before one can see this problem as having been overcome. Certainly there is no fixed time limit, I personally tend to see the names Leibniz and Newton as the beginning of the understood infinity. And with the so-called normal citizen, infinity in the sense of an infinite stringing together that leads to something finite has not yet arrived. Greetings Wauzi \ (\ endgroup \)|
Re: The Zeno Paradoxby: Ex_Mitglied_40174 on: Fri. March 24, 2006 14:01:37
|\ (\ begingroup \) I think the following: If someone says that Archilles HAVE reached the point where the turtle was (at some point), then he compares a spatial (place) point with a temporal (time) point . And, in the final analysis, this is not permitted ............. Completely agree. Michael \ (\ endgroup \)|
Re: The Zeno Paradoxby: Bernhard on: Tue. April 18, 2006 00:10:41
|\ (\ begingroup \) Congratulations! I think it's especially nice that you pointed out that Zeno didn't just set his paradoxes in the air because he thought of a joke that he wanted to get rid of. He wanted to bring examples to show that this, downright "static" consideration of a movement is not enough - even though we fall for it again and again today. In addition, I would like to add two other paradoxes here: The second paradox is a kind of reversal of the first: Before an object, e.g. an arrow has covered half of its trajectory, it must first traverse a quarter of the total path, first an eighth, etc. This consideration can be repeated any number of times and results in an infinite regression - which would show that the arrow not only never arrives, but has never been shot! The third paradox looks at the movement itself: Zenon looks at a moving object, e.g. again an arrow at any point in time in space and says "He is standing!". But if at any given point in time it is equally immobile, how can it fly? While the first two paradoxes already show the key points of infinitesimal calculus, the third with the problem of the "simultaneous" measurement of place, time and movement comes damn close to W. Heisenberg's uncertainty principle. Thanks again for the stimulating contribution, Bernhard \ (\ endgroup \)|
Re: The Zeno Paradoxby: Nodorsk on: Fri. May 19, 2006 10:48:34 pm
|\ (\ begingroup \) Hello, thank you for the positive review;) I'm glad that you like the article ... Greetings \ (\ endgroup \)|
Re: The Zeno Paradoxfrom: Ex_Mitglied_40174 on: Mon. May 22, 2006 22:45:14
|\ (\ begingroup \) Hello, the calculation looks good, but I believe that I found an error in the calculation .... in the following place: Instead of S_n = 1 + q (S_n - q ^ n) S_n - q S_n + q ^ (n + 1) = 1 S_n = (1 - q ^ (n + 1)) / (1 - q). -> lim (n -> \ inf, S) = lim (n -> \ inf, (1 - q ^ (n + 1)) / (1 - q)) = 1 / (1 - q) it should are called S_n = 1 + q (S_n - q ^ n) S_n - q S_n + q ^ (n + 1) = -1 S_n = (q ^ (n + 1) - 1) / (q - 1). and thus lim (n -> \ inf, S) = lim (n -> \ inf, (q ^ (n + 1) - 1) / (q - 1)) = 1 / (1 - q) ie Sn = 1 + q ^ 1 + q ^ 2 + q ^ 3 ... + q ^ n = (q ^ (n + 1) - 1) / (q - 1) =! (1 - q ^ (n + 1)) / (1 - q) there is a sign error. best regards The One \ (\ endgroup \)|
Re: The Zeno Paradoxby: huepfer on: Tue. May 23, 2006 15:59:51
|\ (\ begingroup \) Hello The One, if you multiply both numerator and denominator by (-1) in the original version, you get your version. So the two fractions are identical. Regards, Felix \ (\ endgroup \)|
Re: The Zeno Paradoxfrom: Ex_Mitglied_40174 on: Tue. June 12, 2007 13:53:11
|\ (\ begingroup \) Here the negative time is ignored! The turtle runs slower than Achilles. The speed of S and A are compared and subtracted from each other. Achilles 100 kmh and turtle 10 kmh. So 10 minus 100 equals -90. minus 90 must be built into the calculation. This makes the equation solvable, also as a paradox (or has a solution that is not a paradox)! 10 and 100 are only used as an example, because I don't know how fast Achilles was and how fast the turtle was or is. \ (\ endgroup \)|
Re: The Zeno Paradoxfrom: Ex_Mitglied_40174 on: Tue. January 29, 2008 10:18:02 p.m.
|\ (\ begingroup \) Hello The One, hello Felix! The One writes: ... it should read (1) S_n = 1 + q (S_n - q ^ n) (2) S_n - q S_n + q ^ (n + 1) = -1 (3) S_n = (q ^ (n + 1) - 1) / (q - 1). Here you made 2 mistakes: At (2), 1 and not -1 remains at the end. From your equation (2) it follows: S_n = (-q ^ (n + 1) - 1) / (q - 1) and not the equation (3). One mistake cancels the other and in the end, dear Felix, it's true. It's easy to do in math! Regards, Michael \ (\ endgroup \)|
Re: The Zeno Paradoxby: Ex_Mitglied_40174 on: Thu. January 31, 2008 8:30:24 am
|\ (\ begingroup \) Die - Revenge of Achilles: When Achilles came to the starting line of the turtle, he stopped the run and called Zenon to him. The following dialogue developed: Achilles: What do you see, Zeno? Zeno: I see the turtle is in front of you. Achilles: Right. And what do you see behind me now? Zeno: Oh, there's another turtle. How does it get there? Achilles: This is my house turtle, which I brought with me without your knowledge and positioned it in the middle between me and your turtle and it competed with us. She was in front of me when I started and now you can see her behind me. Can you explain to me now why I was able to overtake one and not the other, although both ran at the same speed? Zenon is still wondering how something like this is possible. Regards, Michael \ (\ endgroup \)|
Re: The Zeno Paradoxfrom: Ex_Mitglied_40174 on: Fri. February 01, 2008 19:47:09
|\ (\ begingroup \) Solution of a pupil in grade 6 Once upon a time there was a grandpa who liked to play with his granddaughter. When his Annalena was around 3 years old, they had a lot of fun running after them. Later, when Annalena had mastered the four basic arithmetic operations, Grandpa gave Annalena the following task: Let us assume that you had a lead of 12 m when running afterwards and that you were always walking straight ahead. Then can you calculate how far you and I will run before I catch up with you if I run 3 times faster than you? After a while Annalena replied: I'll be 6 m and you will run 18 m. When Grandpa asked how she came up with the correct result, she replied: Grandpa, that's really easy, really easy: If I subtract from your running route that is 3 times shorter, 2 of my running routes remain for the lead . So I have to divide 12m by 2 and that makes 6m for me, and disappeared. A task for the next mathematics PISA test for school grade 6 !!! You can therefore calculate the catching-in distance even without knowledge of algebraic or limit values. Regards, Michael \ (\ endgroup \)|
Re: The Zeno Paradoxby: Ex_Mitglied_40174 on: Sun. 03 February 2008 10:08:27
|\ (\ begingroup \) Hello, Nodorsk! If I compare your article with the one from Wikipedia, I notice that the curtain falls on Wikipedia and all questions are answered satisfactorily, while you still have questions unanswered. I agree and I also like your presentation very much. I also asked myself the question: How long (in terms of time) do the two run until they meet each other? The answer I found was only a certain length of the hauling in distance of Achilles and the turtle, whereas their common running time remains indefinite. In your solution for a student, the last term for the retrieval path as well as for the retrieval time contains two unknown quantities. These are the speeds of the two runners. These are unknown: For Zenon there is only one fast, but not one fast! You can divide the numerator and denominator by vs in the term of the recovery path and thus get S = (W * k) / (k-1), where k is the ratio of the two speeds. That gives a certain numerical value. Unfortunately, this is not possible with the recovery time. The recovery time remains indefinite: [T = W / (vs * (k-1))]. How big it is depends on the speed at which Achilles (and then the turtle) runs. There are so many mathematical (theoretical) answers for the catch-up time. The presented mathematical-physical solution methods result in an infinite number of solutions. In reality there is only one solution: Achilles will run at the highest possible speed (he will move his legs with the greatest frequency and with the greatest stride length or with a compromise between these). Conclusion: There is also the possibility of in the so-called mathematical-physical solutions. . ., and so on, of going on like this. The feature of the infinite is thus not eliminated, it only expresses itself in a different form: one can always give another solution to a solution (a pair of quantities path-time). Therefore one can claim that the paradox has not been solved so mathematically and physically. Zenon is right (with his assumptions). Zeno conducts a thought experiment to predict something. To do this, he needs a certain amount of information, for him that is the spatial advantage and the ratio of the two speeds. How does he actually get the opportunity to continue? From the purely mathematical process (with the basic calculations or the limit value of a mathematical series) one will not come across a saving thought, but from the algebraic-physical process: A third statement is missing. A spatial advance always includes a temporal advance (since Einstein-Minkovski it has been said: There is no time or space per se, there is only a spatial time or a temporal space). In the race, we refer to the starting line of Achilles. If you give the turtle a head start, it actually has to get there first. But for this it needs a very specific time (also in relation to the starting line of Achilles) and you have to specify this (this is how you know the speed of the turtle). The possibility of the infinite can no longer appear like this: You only get a single, unambiguous solution. A possible explanation why the ancient Greeks could not solve the paradox: The ancient Greeks did not know the term fraction, they only spoke of relationships, i.e. H. they compared one time to another and one way to another. The time and the way to combine them into a unit would have been inconceivable, unacceptable for them. Only after hundreds of years, when the fractions were a matter of course, could the linear, uniform movement be fully described with the physical concept of speed (since then there has been the unit space-time or time-space, we are only really aware of this in our time). Zenon's merit lies in the fact that he found a gap in the thought world of his time, but unfortunately he misinterpreted it: He was just a child of his time, just as we are children of our time. I will come back to your pictures (final conclusion). Regards, Michael \ (\ endgroup \)|
Re: The Zeno Paradoxby: Ex_Mitglied_40174 on: Tue. February 05, 2008 10:47:20
|\ (\ begingroup \) Hello, Nodorsk (continuation and end), how right Wittgenstein is. Actually, everyone has long known: theory and practice (experienced reality) are two different things. Every theory is a combination of concepts, which in turn are abstractions. Every abstraction is an ignoring, more or less features of what I mean are always being ignored. The higher the abstraction, the fewer features the term has, but on the other hand it includes more things quantitatively. The term point in mathematics and theoretical physics includes everything that can move. Achilles and the turtle are seen as points. But also the places where you are right now, and of course the time too. But what are time points, body points, place points? Nothing but abstractions at the highest level. And now to your two pictures of Achilles and the turtle. These are two-dimensional representations of three-dimensional creatures, so something is missing. The mathematical-physical solution goes even further and makes it zero-dimensional, i.e. H. ignores all dimensions. Now that may sometimes work, but not with living creatures, because they move by deforming themselves, changing their dimensions. That brings me to Plato, whom you bring into play, in context, I thank you for that. We speak of Achilles in the singular but differentiate between outer and inner parts, such as legs, arms, head or heart, nervous system, etc. (I don't even want to go to the molecules, etc.), i.e. H. Achilles is not only one thing for us, but also many things, only I cannot think both at the same time. I can only concentrate consciously on one aspect (I can only run after one and not two who run in different directions). Now with Achilles and the turtle, the parts that we distinguish in them do not move in the same way during locomotion: the legs move differently than the arms and differently than the head, to name but a few. If I only move my arms, I will not be able to step forward while standing on my feet. The legs are the deciding factor for locomotion. However, if I only move one leg, I am not yet advanced either, first the unit is moved forward by one foot, followed by the pulling of the second, results in a progressive movement of the Achilles unit. In other words: Achilles cannot move from point in time to point in time but only ever in a time interval (which is not a point in time): It is pointless (at least in this case) to continue dividing time without end (to choose a smaller time unit). But now I have to call it a day. If you have the time and inclination, you can spin the thread further in order to come to a new conclusion. What astonishes and fascinates me again and again is the fact that there are so many perspectives (from which I always learn something new) for such a short and so simple, if not to say banal, story (Achilles just has to walk there, where the turtle will be, and not where it was, even if this goes without end. Among these there are those who achieve the set goal, but lead others astray. Visit http://de.wikipedia.org/wiki/Benutzer:Istvancsek Regards, Michael \ (\ endgroup \)|