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Zeno’s Paradoxes or What Happened When Achilles and the Hare Decided to Outfox the Legendary Tortoise

A drawing showing Greek champion Achilles chasing the hare and the tortoise in a race to the finish, with the legendary tortoise in the lead.Wacky Races

How could the humble Tortoise ever beat legendary Greek champion, Achilles, in a race to the finish?  And what about that time when the champion of the animal kingdom simply ridiculed his next-door neighbour aka the Hare?

The old story concerns a hare who ridicules a slow-moving tortoise and is challenged by him to a race.  Confident of winning, the hare soon leaves the tortoise behind… and even decides to take a nap midway through the course.  When he awakes, however, he finds that his competitor, crawling slowly but steadily, has arrived before him.  The moral of the story could be that slow and steady is more effective in the long run.

Frankly, the Hare had it coming for being so cocky and taking a nap half-way through the race!

Tortoise 1 – Hare 0

 

Zeno of Elea

Greek philosopher Zeno (c. 490 – c. 430 BC) came up with this interesting apparent paradox, also involving the Tortoise.  In Zeno’s most famous paradox, Achilles gets involved in a footrace with the tortoise. 

A diagram explaining Zeno's famous paradox of Achilles and the tortoise. The tortoise is the first to start racing, closely followed by Achilles. As Achilles must cover at least half the distance that separates him from the tortoise, it follows that theoretically-speaking Achilles can never catch up to the tortoise.
In Zeno’s Paradox, Achilles the Champion finds it impossible to outrun its competitor, the famous Tortoise.

Achilles allows the tortoise a head start of 100 metres, for example.  If we suppose that each racer starts running at some constant speed (one being very fast, and the other one being very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise’s starting point.

During this time, the tortoise has run a much shorter distance, say, 10 metres.  Then, it will take Achilles further time to run that distance, by which time the tortoise will have advanced farther, and then Achilles needs more time still to reach this third point, while the tortoise continues to move ahead.

Whenever Achilles reaches somewhere the tortoise has been, he still has farther to go.  Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, the Greek hero can never overtake the tortoise.

Tortoise 1 – Greek Hero “Nil Points”

 

A diagram illustrating another one of Zeno's paradoxes - the one about the arrow that never reaches its target, because once again it has to travel each time at least half the distance that separates it from its intended target.Zeno also states that for motion to occur, an object must change the position which it occupies.  He gives the example of an arrow in flight.  He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not.  It does not move to where it is not, because no time elapses for it to move there.  It cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then…

Motion is impossible!

 

Well, clearly, yes…  The logic is flawless, but this is taking things to extreme.  In the paradox, Achilles can never actually get anywhere – let alone overtake the tortoise – since the distance he would have to cover could be halved an infinite number of times: halfway there, halfway again, and so on.  Therefore, Achilles would have to take an infinite number of ever-smaller steps to reach his goal.

Experience of the real world does tell us otherwise.

 

Solving The Dichotomy

Look, it’s like this!

Modern science can resolve Zeno’s paradoxes.

Before an object can travel a given distance d, it must first travel a distance \frac{d}{2}.

In order to travel \frac{d}{2}, it must travel \frac{d}{4}, and so on and so forth…

Since this sequence goes on forever, it appears that the distance d cannot be travelled.

A humoristic meme showing perhaps the only way for the rabbit of beating Zeno's paradox. The photograph shows a bunny rabbit riding on the back of a small pet turtle.
An alternative way of resolving Zeno’s paradox…

The resolution of the paradox awaited for the invention of calculus and the proof that infinite geometric series such as 

\sum ^{\infty} _{i=1} {(1/2)}^i =1

can indeed converge, so that the infinite number of “half-steps” needed is balanced by the increasingly short amount of time needed to travel those distances.

Voilà!