┴———————┴————┴———┴——┴——┴——
A B C D E F……
about the question:When Achilles is at A, the tortoise is at B, when he catches to B, the tortoise is a C and so on....
And as we can see, the Achilles is closer and closer to the tortoise, and the distance is smaller and smaller, every distance is half of the distance before, but it will never be 0 so Achilles will never catch the tortoise.
The reason of this is because Zeno ignore a very important thing:because the distance is smaller, the time Achilles uses for finishing the distance will also be smaller,and always the half of the time before. Zeno in fact uses 2 different measures of time.We called the time Zeno use the "Zeno time"to differ it from the real time. And we measure the time by how many times Achilles catches the tortoise. For example, if Achilles arrives n times of the n start point of the tortoise, the Zeno time is n. So if Zeno time is finite, Achilles can never catch the tortoise.Zeno time cannot measure the time after Achilles catch the tortoise. But in our real time if he uses 1 minute to finish the AB, then he will use 30 seconds to finish the BC and 15 seconds to finish CD and so on...So the time can be calculate.
ReplyDeleteThis is an excellent distinction Vivien, but you are also making a controversial assumption: that there is an infinite amount of time within which Achilles can finish the race. Isn't this a problem in itself? Is it valid to introduce this assumption without proof and do we need such a strong assumption to solve the problem?