Achilles and the Tortoise

阿基里斯與烏龜

How could a humble tortoise

一隻不起眼的烏龜

beat the legendary Greek hero Achilles in a race?

怎樣才能在賽跑中戰勝希臘英雄阿基里斯呢？

The Greek philosopher Zeno liked a challenge

希臘哲學家齊諾喜歡接受挑戰

and came up with this paradox

提出了以下悖論

First, the tortoise is given a slight head start

首先允許烏龜先跑一段距離

Anyone fancying a flutter

喜歡打賭的人

would still rush to put their money on Achilles

還是會把他們的寶壓在阿基里斯身上

But Zeno pointed out that

但齊諾指出

to overtake him

要趕上烏龜

Achilles would first have to cover the distance

阿基里斯首先必須跑到

to the point where the tortoise began

烏龜和他共同起跑時的起跑點上

In that time the tortoise would have moved

而在這個時間段烏龜又往前跑了一段距離

so Achilles would have to cover that distance

所以阿基里斯還得跑完烏龜跑完的第二段距離

giving the tortoise time to amble forwards a bit more

在這段時間烏龜又向前爬了一截

Logically this would carry on forever

從邏輯上看這樣的過程會一直持續下去

However small the gap between them

無論他們之間的差距多麼微小

the tortoise would still be able to move forwards

在阿基里斯追趕的時間

while Achilles was catching up

烏龜總是能夠向前爬一段距離

Meaning that Achilles could never overtake

這就是說阿基里斯永遠無法趕上烏龜

Taken to an extreme

更極端的情況是

this bizarre paradox suggests

這個奇怪的悖論顯示

that all movement is impossible

所有的運動都是不可能的

But it did lead to the realization

但卻說明了一個理論

that something finite can be divided

即有限的事物

an infinite number of times

總是能被無限次的分割

This concept of 'an Infinite Series' is used in finance

“無限序列”的概念被應用到了金融業

to work out mortgage payments...

用來計算人們的按揭款

Which is why they take

這就是為什麼無論償還多少次

an infinite amount of time to pay off!

他們都無法還清債務的原因