# Can we prove the Zenos paradox to be false?

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?

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."
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 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 \)