What is Zeno's Dichotomy Paradox? - Colm Kelleher

什麼是芝諾的二分法悖論？ - Colm Kelleher

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

譯者: Jephian Lin 審譯者: Regina Chu

This is Zeno of Elea,

這位是埃利亞的芝諾，

an ancient Greek philosopher

一位古希臘哲學家，

famous for inventing a number of paradoxes,

因發明許多悖論而聞名。

arguments that seem logical,

悖論是指看似有道理，

but whose conclusion is absurd or contradictory.

但結論卻是荒謬 或矛盾的論證。

For more than 2,000 years,

兩千多年以來，

Zeno's mind-bending riddles have inspired

芝諾那些誤導思路的難題 啟發了許多

mathematicians and philosophers

數學家與哲學家

to better understand the nature of infinity.

來了解「無窮」的本質。

One of the best known of Zeno's problems

最有名的一個芝諾難題

is called the dichotomy paradox,

叫做二分法悖論，

which means, "the paradox of cutting in two" in ancient Greek.

在古希臘文的意思就是「切割為二詭辯」。

It goes something like this:

內容大約是這樣：

After a long day of sitting around, thinking,

芝諾在漫長地坐著沉思一天後，

Zeno decides to walk from his house to the park.

決定從家裡散步到公園。

The fresh air clears his mind

清新的空氣啟發他的心靈

and help him think better.

並讓他想得更清楚。

In order to get to the park,

要走到公園，

he first has to get half way to the park.

他必須先走到路程的中點。

This portion of his journey

他這部份的旅程

takes some finite amount of time.

要花一些有限的時間。

Once he gets to the halfway point,

一旦他到達這中點，

he needs to walk half the remaining distance.

他必須再走到剩下距離的中點。

Again, this takes a finite amount of time.

這又花了一些有限的時間。

Once he gets there, he still needs to walk

一旦他到那兒，他還是必須再走到

half the distance that's left,

剩下距離的中點，

which takes another finite amount of time.

那也會花另一些有限的時間。

This happens again and again and again.

這會一次又一次的發生。

You can see that we can keep going like this forever,

你可以見到我們 永遠都在這過程打轉，

dividing whatever distance is left

就是不斷將剩的距離分成

into smaller and smaller pieces,

更小更細的路段，

each of which takes some finite time to traverse.

每一段都須要一些 有限的時間才能通過。

So, how long does it take Zeno to get to the park?

所以，芝諾要多久才能走到公園？

Well, to find out, you need to add the times

嗯，要得到答案，你必須把每段路段

of each of the pieces of the journey.

所花的時間加起來。

The problem is, there are infinitely many of these finite-sized pieces.

而問題是，有無限個這種 「有限的時間」。

So, shouldn't the total time be infinity?

所以，全部的時間也應該要是無限大嗎？

This argument, by the way, is completely general.

順帶一提，這個論證是很通用的。

It says that traveling from any location to any other location

它說明從任何地點移動到 任何其它地點

should take an infinite amount of time.

應該要花無窮的時間。

In other words, it says that all motion is impossible.

換句話說，它說明所有運動都是不可能的。

This conclusion is clearly absurd,

這個結果顯然很荒謬，

but where is the flaw in the logic?

但邏輯上的瑕疵在哪裡？

To resolve the paradox,

要解開這個悖論，

it helps to turn the story into a math problem.

把故事轉換成數學問題 會有所幫助。

Let's supposed that Zeno's house is one mile from the park

我們假設芝諾的家 距離公園有一英里，

and that Zeno walks at one mile per hour.

而芝諾每小時走一英里。

Common sense tells us that the time for the journey

常理告訴我們這趟旅程

should be one hour.

應該要花一小時。

But, let's look at things from Zeno's point of view

但是，讓我們從芝諾的觀點來看看

and divide up the journey into pieces.

並把路程分程許多小段。

The first half of the journey takes half an hour,

最初的一段路程要花 1/2 小時，

the next part takes quarter of an hour,

下一段要花 1/4 小時，

the third part takes an eighth of an hour,

而第三段要花 1/8 小時，

and so on.

Summing up all these times,

以此類推。

將這些時間全部加起來，

we get a series that looks like this.

我們得到一串 長成這樣的級數。

"Now", Zeno might say,

「現在」，芝諾可能會說，

"since there are infinitely many of terms

「因為方程式右邊有無限項，

on the right side of the equation,

每項又都是有限的，

and each individual term is finite,

它們的總和

the sum should equal infinity, right?"

應該是無窮，對吧？」

This is the problem with Zeno's argument.

這就是芝諾論證的問題了。

As mathematicians have since realized,

數學家從此明白，

it is possible to add up infinitely many finite-sized terms

把無限個有限的量相加 是有可能得到

and still get a finite answer.

一個有限的答案。

"How?" you ask.

「怎麼會呢？」你可能會問。

Well, let's think of it this way.

嗯，我們可以這樣想。

Let's start with a square that has area of one meter.

我們考慮一個 一公尺見方的正方形。

Now let's chop the square in half,

現在把這個正方形 分成兩半，

and then chop the remaining half in half,

再把剩的分半，

and so on.

接著往下做。

While we're doing this,

當我們這麼做時，

let's keep track of the areas of the pieces.

我們依序記錄每塊的面積。

The first slice makes two parts,

最初的切片有兩部份，

each with an area of one-half

每部份的面積都是 1/2，

The next slice divides one of those halves in half,

而下一次切片把其中一個 1/2 再分成兩半，

and so on.

依此類推。

But, no matter how many times we slice up the boxes,

但，無論我們切割了幾次，

the total area is still the sum of the areas of all the pieces.

整塊面積還是所有小面積的總和。

Now you can see why we choose this particular way

現在你可以了解 為什麼要選這麼特別的方式

of cutting up the square.

來切割正方形。

We've obtained the same infinite series

我們已經做出了那串 相同的無窮級數，

as we had for the time of Zeno's journey.

就是在芝諾的旅程中 算出來的那串。

As we construct more and more blue pieces,

當我們建構了更多的藍色小方塊，

to use the math jargon,

用數學的行話來說，

as we take the limit as n tends to infinity,

就是當我們取 n 趨近到無窮時的極限，

the entire square becomes covered with blue.

整個正方形都被藍色蓋住了。

But the area of the square is just one unit,

但正方形的面積就只有 1 平方單位而已，

and so the infinite sum must equal one.

所以無窮項的總合一定是 1。

Going back to Zeno's journey,

我們回到芝諾的旅程，

we can now see how how the paradox is resolved.

我們可以看到這悖論 是如何被解決的。

Not only does the infinite series sum to a finite answer,

不止是無限項加起來可能是有限，

but that finite answer is the same one

而且這個有限的答案還是一樣的，

that common sense tells us is true.

和常理告訴我們的一樣 ──

Zeno's journey takes one hour.

芝諾的旅程要花一小時。