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Zeno’s Paradox – Achilles and the Tortoise
This is a very famous paradox from the Greek philosopher Zeno – who argued that a runner (Achilles) who constantly halved the distance between himself and a tortoise would never actually catch the tortoise. The video above explains the concept.
There are two slightly different versions to this paradox. The first version has the tortoise as stationary, and Achilles as constantly halving the distance, but never reaching the tortoise (technically this is called the dichotomy paradox). The second version is where Achilles always manages to run to the point where the tortoise was previously, but by the time he reaches that point the tortoise has moved a little bit further away.
Dichotomy Paradox
The first version we can think of as follows:
Say the tortoise is 2 metres away from Achilles. Initially Achilles halves this distance by travelling 1 metre. He halves this distance again by travelling a further 1/2 metre. Halving again he is now 1/4 metres away. This process is infinite, and so Zeno argued that in a finite length of time you would never actually reach the tortoise. Mathematically we can express this idea as an infinite summation of the distances travelled each time:
1 + 1/2 + 1/4 + 1/8 …
Now, this is actually a geometric series – which has first term a = 1 and common ratio r = 1/2. Therefore we can use the infinite summation formula for a geometric series (which was derived about 2000 years after Zeno!):
sum = a/(1-r)
sum = 1/(1-0.5)
sum = 2
This shows that the summation does in fact converge – and so Achilles would actually reach the tortoise that remained 2 metres away. There is still however something of a sleight of hand being employed here however – given an infinite length of time we have shown that Achilles would reach the tortoise, but what about reaching the tortoise in a finite length of time? Well, as the distances get ever smaller, the time required to traverse them also gets ever closer to zero, so we can say that as the distance converges to 2 metres, the time taken will also converge to a finite number.
There is an alternative method to showing that this is a convergent series:
S = 1+ 1/2 + 1/4 + 1/8 + 1/16 + …
0.5S = 1/2+ 1/4 + 1/8 + 1/16 + …
S – 0.5S = 1
0.5S = 1
S = 2
Here we notice that in doing S – 0.5S all the terms will cancel out except the first one.
Achilles and the Tortoise
The second version also makes use of geometric series. If we say that the tortoise has been given a 10 m head start, and that whilst the tortoise runs at 1 m/s, Achilles runs at 10 m/s, we can try to calculate when Achilles would catch the tortoise. So in the first instance, Achilles runs to where the tortoise was (10 metres away). But because the tortoise runs at 1/10th the speed of Achilles, he is now a further 1m away. So, in the second instance, Achilles now runs to where the tortoise now is (a further 1 metre). But the tortoise has now moved 0.1 metres further away. And so on to infinity.
This is represented by a geometric series:
10 + 1 + 0.1 + 0.01 …
Which has first time a = 10 and common ratio r = 0.1. So using the same formula as before:
sum = a/(1-r)
sum = 10/(1-0.1)
sum = 11.11m
So, again we can show that because this geometric series converges to a finite value (11.11), then after a finite time Achilles will indeed catch the tortoise (11.11m away from where Achilles started from).
We often think of mathematics and philosophy as completely distinct subjects – one based on empirical measurement, the other on thought processes – but back in the day of the Greeks there was no such distinction. The resolution of Zeno’s paradox by use of calculus and limits to infinity some 2000 years after it was first posed is a nice reminder of the power of mathematics in solving problems across a wide range of disciplines.
The Chess Board Problem
The chess board problem is nothing to do with Zeno (it was first recorded about 1000 years ago) but is nevertheless another interesting example of the power of geometric series. It is explained in the video above. If I put 1 grain of rice on the first square of a chess board, 2 grains of rice on the second square, 4 grains on the third square, how much rice in total will be on the chess board by the time I finish the 64th square?
The mathematical series will be:
1+ 2 + 4 + 8 + 16 +……
So a = 1 and r = 2
Sum = a(1-r64)/(1-r)
Sum = (1-264)/(1-2)
Sum = 264 -1
Sum = 18, 446,744, 073, 709, 551, 615
This is such a large number that, if stretched from end to end the rice would reach all the way to the star Alpha Centura and back 2 times.
Essential Resources for IB Teachers
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Essential Resources for both IB teachers and IB students
1) Exploration Guides and Paper 3 Resources
I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission. Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator! I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
IB Maths Resources from Intermathematics 
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August 26, 2021 at 10:34 am
Giuseppe Gori
When two procedures used to solve a problem lead to different results (a paradox), the solution to the paradox is to show why one of the procedures is wrong. It is NOT showing why the wrong procedure must, would, or should lead to the same result. A complete explanation, and the solution, are given below.
What Does Solving a Paradox Mean?
As mentioned at the beginning of my article (at: https://bit.ly/2IM76rF ), a paradox proposes the existence of two different results as a solutions for the same problem. These results are inconsistent with each other, depending on which procedure is used. Only one can be correct. As Brown and Moorcroft suggest, we are not looking for a mathematical demonstration that Achilles reaches the tortoise. Assuming they are both running in the same direction, we know he will. We can calculate the exact time, given the distance between the two and the two speeds, using a simple formula:
t = distance/(difference in velocity)
Instead, explaining or invalidating a paradox is to show a fault in the paradox formulation, or the proposed solving procedure, so that we can exclude this procedure and demonstrate that there is only one result for the original problem. The solution of a paradox is the answer to the question: “How does the paradox formulation misrepresent reality or logic?” That is, we need to show why the proposed method is conceptually wrong. Solving a paradox, invalidates the formulation of a problem proposed by the author of the paradox and leaves us with only valid procedures for solving the original problem.
Why were the previous proposed solutions for the paradox not satisfactory?
The fault of most, if not all, the proposed solutions to Zeno’s paradox is the assumption that Zeno’s proposed procedure was correct. The procedure seems to be logical when it is first introduced to us, but we will see that the procedure proposed by Zeno is conceptually wrong. The authors then tried using Zeno’s faulty procedure to reach the expected correct result for the original problem. This is unacceptable.
The Explanation of Zeno’s Paradox:
Zeno’s proposition invites the solver to do a series of steps each time changing system of reference: STEP 1: The starting system of reference: The point where Achilles starts the race and the tortoise is well ahead, STEP 2: After a while, we are then asked to use a new system of reference: The point where Achilles reached and where the tortoise initially started, with the the tortoise now a bit further ahead, STEP 3: Then again we are asked to use, recursively, a new system of reference with the new starting point for Achilles and with the tortoise still further ahead, with every step we are asked to freeze the process and then continue by re-creating and examining the original problem using a different system of reference.
Today we know more about the relative motion of two bodies. Solving a problem that involves space and time, requires a defined system of reference, which cannot be changed without the proper conversions. After Zeno’s proposed first step, or first change of system of reference, the problem, as presented in the second step, is exactly the same as the original, the only change being a difference in “scale”. No progress was achieved in solving the problem. Changing system of reference essentially restarts the problem-solving procedure. This realization implies that the problem is never going to reach a conclusion as the step by step procedure is reiterated. If the system of reference is changed at every step, our working spacetime shrinks with every step, the solution becomes elusive and the tortoise becomes apparently unreachable. Zeno proposes a procedure that never ends, for solving a problem that has a trivial solution.
A programming analogy
Zeno’s proposed procedure is analogous to solving a problem by recursion, a well known problem solving technique available in modern programming languages. However, recursion would be the wrong technique to solve the original problem. If the problem was programmed exactly as Zeno suggested, the program would never end normally, simply because the condition for the end of the recursion process (Achilles reaches the tortoise) would never occur. Our computer program would confirm the paradox!
Our Solution
Our solution of Zeno’s paradox can be summarized by the following statement:
“Zeno proposes observing the race only up to a certain point, using a system of reference, and then he asks us to stop and restart observing the race using a different system of reference. This implies that the problem is now equivalent to the original and necessarily implies that the proposed procedure for solving the problem will never end.”
That’s it. You cannot change system of reference in the middle of a problem involving space and time, whether the system of reference is openly stated, or implied. As an analogy, you cannot solve a problem involving measurements by using English Imperial measures at the start of calculations and then switch to metric measures (without proper conversions) in the middle of calculations.
An example
The following is not a “solution” of the paradox, but an example showing the difference it makes, when we solve the problem without changing the system of reference. In this example, the problem is formulated as closely as possible to Zeno’s formulation.
Zeno would agree that Achilles makes longer steps than the tortoise. Let’s assume that one Achilles-step is about 20 tortoise-steps long, and let’s also assume that both Achilles and the tortoise make the same number of steps in the same amount of time. For example, two steps per second (the exact amount doesn’t really matter). If the tortoise starts the race 20 Achilles-steps ahead of him, then after 20 steps Achilles reaches where the tortoise was (See diagram below: Tortoise starting point).
(The picture can be seen at: https://bit.ly/2IM76rF )
In the meantime, the tortoise has made 20 of her steps, and she is now one full Achilles-step ahead of him. We have not changed our system of reference. We referred to both starting points. These did not move relatively to each other. We could choose any fixed ground point. To please Zeno, let’s continue by referring to the tortoise starting point, where Achilles currently is. When both runners make one more step, step 21, the tortoise will have moved by one of her steps and she will still be ahead of Achilles by that one tortoise-step. Achilles is now one Achilles-step ahead of the tortoise starting point. Now, let’s continue, without changing the system of reference. This is the key point. We do not redefine the problem and use the current positions of the runners as new starting points, as Zeno proposes, but we refer to the information about the race we have already accumulated in our knowledge base. Achilles then completes his 22nd step, and he is two Achilles-steps ahead of the tortoise starting point. The tortoise will have completed her 22nd tortoise-step from her starting point. Hence the tortoise is now behind Achilles by 18 tortoise-steps. Thus, if we do not change the system of reference, the paradox does not appear.