#### Achilles and the Tortoise

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

This is what’s said in Aristotle Physics (VI:9, 239b15) about the paradox. The paradox goes like this: Achilles is in a foot race with a tortoise.

As seen that the tortoise  is probably much slower than Achilles, he allows the tortoise a head start of say 200 metres.

Then the race starts, and we suppose both of them start running at a constant speed (one very fast, one very slow). Within a certain period of time, Achilles will have come to that 200 meter point where the tortoise started. But within the same period, the tortoise will have already come for example 20

metres further. Then again it takes some time for Achilles to come 20 metres further, but at the same time the tortoise will have moved on 2 metres. Then again, it will take Achilles some time to get 2 metres further, while within that time the tortoise will have moved 20 cm further. This can go on and on, and will never stop, meaning Achilles will never overtake the tortoise, because when he’s at the point where the tortoise was some time ago, he still has further to go. Therefore, motion is nothing but an illusion.

That which is in locomotion must arrive at the half-way stage before it arrives at the goal. - Aristotle Physics (VI:9, 239b10)

Suppose you wanted to cycle to school or work. Before you could get there, you must get half-way to school, right? But before you could possibly get there, you need to cycle half-way that distance (which is quarter-way the complete distance). But, you must arrive at half-way that distance before you could get there. And so on and on. Written in numbers to make things clear: To get from A to B, you must first get to 0.5AB. But, before you can get there, you will need to arrive at 0.25AB. But, to get there, you must arrive at 0.125AB.

This requires an infinite amount of tasks to be done before you could arrive at your goal, which Zeno maintains is an impossibility.

And next to that: there’s no first distance to be run, as seen that every finite first step, could always be divided in half. Therefore, the movement from A to B can not be begun. And, if you can’t start going towards a goal, can you ever get there? No. Therefore, motion is nothing but an illusion.

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. -Aristotle Physics (VI:9, 239b5)

This declaration is quite cryptic, so let me explain. Zeno states that, for motion to occur, one object needs to switch the space or position it occupies. If you want to get from A to B, you’ll need to change the position you occupy from A to B. Zeno takes an arrow in flight as example. At any instant of time, an object can only be moving to where it is or where it is not for motion to occur, is a logical conclusion. But, at any instant of time, the arrow is neither moving to where it is as to where it is not. As seen that it can’t move to where it already is, and there’s no time to elapse to where it is not (as it is an instant of time). As seen that time consists completely of a lot of instants, and we just concluded that everything is motionless at every instant, motion is nothing but an illusion.

Though all three paradoxes look like each other, the first two divide space into segments, but the last one divides time into instants.

#### Conclusion

Well, there isn’t really much to conclude. Zeno made some pretty good paradoxes, which are not easy to contradict, but are at the same time quite simple, which make them so famous. Feel free to leave a comment  with another paradox or that very well explains why a paradox isn’t legitimate!

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### One Response to The Paradox(es) of Zeno

1. Ardash K. says:

Here’s our analysis of this paradox. It describes how you’re looking at the race, not how the race actually happens. If you consider the normal flow of time, the paradox is invalid. Details: