### Solution to Zeno's Second Paradox

Zeno’s Second Paradox purports to show that in a chase, the chaser can never catch the person or object being chased. In the paradox the Greek hero Achilles graciously gives his slower rival, which legend has turned into ‘The Tortoise’, a head start.

At the start of the race, Achilles is at position 0 while the tortoise is at the 1 kilometre mark. For convenience, imagine that the tortoise can only run half as fast as Achilles.

Common sense may lead you to think that Achilles would overtake the tortoise after running 2 kilometres. However, by the time Achilles reaches the tortoise’s starting point at the 1 km mark, the tortoise has travelled ½ km ahead, which is 1 + ½ km from Achilles’ starting point at position 0. When Achilles reaches the 1½ km point, the tortoise has reached 1 + ½ + ¼ km, and so on.

When, after N steps, Achilles reaches a distance 2 – (½ )N-1 from the start the tortoise is still in the lead because it is at a distance 2 – (½ )N+1 from the start. No matter how big N (the number of divisions of the journey) becomes, Achilles never overtakes the tortoise!

The solution I propose involves the observation that the pursuer and the pursued are finite entities. (As such is substantially similar to my resolution of Zeno’s First Paradox.)

Say Achilles has a stride length of 2 metres, and the tortoise has a stride length of 1 metre – this tortoise likes to bound along rather than waddle. For convenience, let’s give the tortoise a 10 metre head start. All else being the same (i.e. Achilles and the tortoise takes the same time to make each stride), after 5 strides Achilles is at the 1 km mark, and the tortoise is at the 1.5 km mark. After another 5 strides Achilles will catch up to the tortoise at the 2 km mark.

The crucial thing here is that both Achilles and the tortoise have a finite stride length that remains constant throughout the race. In Zeno’s footrace on the other hand, Achilles and the tortoise make smaller and smaller strides, so that near the 2 km mark they are making infinitessimally small strides. In fact, Achilles never catches up to the tortoise in the Zeno’s Paradox because neither of them ever reaches the 2 km mark. (If you imagine Achilles running the race by himself, you get a situation that is identical to Zeno’s First Paradox.)

The upshot of comparing these two descriptions of the race is that for Zeno’s Second Paradox to be true, both the pursuer and the pursued must be able to reduce their ‘stride length’ an infinite number of times. The only thing that fits this condition is a point (or a line that runs perpendicular to the direction of the race track).

Finite objects such as people and tortoises can reduce their stride length, but not any smaller than the size of their bodies. So the number of reductions they can achieve is finite.

Assuming that there can be no object of a size between a point and an object with finite size, what this means is that when a finite object moves, it does so in finite increments, and there is a minimum finite amount of movement for each finite object.

One question which springs to mind is whether there is a smallest finite amount of movement.

Another is whether there is such a thing as a smallest finite anything. While it’s true that a point, being infinitessimal, is smaller than all finite objects, this does not answer the question of whether there is a smallest among finite objects.

Stimulus: "The Infinite Book", by John D. Barrow. The description of Zero's Second Paradox is largely drawn from pages 21-22 of Barrow's book and also from Zeno's Paradoxes at Wikipedia.com.

2006-03-22 11:48:00am (GMT +10 -1) kyy

At the start of the race, Achilles is at position 0 while the tortoise is at the 1 kilometre mark. For convenience, imagine that the tortoise can only run half as fast as Achilles.

Common sense may lead you to think that Achilles would overtake the tortoise after running 2 kilometres. However, by the time Achilles reaches the tortoise’s starting point at the 1 km mark, the tortoise has travelled ½ km ahead, which is 1 + ½ km from Achilles’ starting point at position 0. When Achilles reaches the 1½ km point, the tortoise has reached 1 + ½ + ¼ km, and so on.

When, after N steps, Achilles reaches a distance 2 – (½ )N-1 from the start the tortoise is still in the lead because it is at a distance 2 – (½ )N+1 from the start. No matter how big N (the number of divisions of the journey) becomes, Achilles never overtakes the tortoise!

**The Solution**The solution I propose involves the observation that the pursuer and the pursued are finite entities. (As such is substantially similar to my resolution of Zeno’s First Paradox.)

Say Achilles has a stride length of 2 metres, and the tortoise has a stride length of 1 metre – this tortoise likes to bound along rather than waddle. For convenience, let’s give the tortoise a 10 metre head start. All else being the same (i.e. Achilles and the tortoise takes the same time to make each stride), after 5 strides Achilles is at the 1 km mark, and the tortoise is at the 1.5 km mark. After another 5 strides Achilles will catch up to the tortoise at the 2 km mark.

The crucial thing here is that both Achilles and the tortoise have a finite stride length that remains constant throughout the race. In Zeno’s footrace on the other hand, Achilles and the tortoise make smaller and smaller strides, so that near the 2 km mark they are making infinitessimally small strides. In fact, Achilles never catches up to the tortoise in the Zeno’s Paradox because neither of them ever reaches the 2 km mark. (If you imagine Achilles running the race by himself, you get a situation that is identical to Zeno’s First Paradox.)

The upshot of comparing these two descriptions of the race is that for Zeno’s Second Paradox to be true, both the pursuer and the pursued must be able to reduce their ‘stride length’ an infinite number of times. The only thing that fits this condition is a point (or a line that runs perpendicular to the direction of the race track).

Finite objects such as people and tortoises can reduce their stride length, but not any smaller than the size of their bodies. So the number of reductions they can achieve is finite.

Assuming that there can be no object of a size between a point and an object with finite size, what this means is that when a finite object moves, it does so in finite increments, and there is a minimum finite amount of movement for each finite object.

**Further questions**

One question which springs to mind is whether there is a smallest finite amount of movement.

Another is whether there is such a thing as a smallest finite anything. While it’s true that a point, being infinitessimal, is smaller than all finite objects, this does not answer the question of whether there is a smallest among finite objects.

Stimulus: "The Infinite Book", by John D. Barrow. The description of Zero's Second Paradox is largely drawn from pages 21-22 of Barrow's book and also from Zeno's Paradoxes at Wikipedia.com.

2006-03-22 11:48:00am (GMT +10 -1) kyy

## 2 Comments:

Thought I'd let you know I've linked to this post from my blog here

Hi, your thoughts are interesting as it is on the right path. However, using the argument for the observed finitary property of physical objects isn't a proper method of deducing the invalidity of Zeno. There are assumptions, such as, your eyes are giving you true information about that scene of the race, the finitude of physical objects is also an assumption. I like that idea, because it gives us a heuristic or observational/"common-sense" reason to reject the indivisibility that Zeno is proposing. However, in order that your proof or solution be complete, it must be totally rigorous. Also, we could change the problem into:

Consider if Zeno thought of not physical athletes and tortoises, but points trying to catch up with another point. I.e if we imagined Achilles as point A, and tortoise point B. Then we reframe the problem, into:

Achilles is at point A, and gives point B a 1 m head start. Now point A moves at such and such a speed, and the tortoise as well. Thus, without considering strides or finitude of objects, we consider whether the process in itself. I.e of the continual need for point A to meet point B, an infinite number of times. I recommend an abstracter way of looking at the problem, due to the philosophical problems due to the inductive nature of observation and our senses in general.

What do you think about what i've said?

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