And why is it so important in mathematics?

而為什麼證明在數學中是如此重要？

Proofs provide a solid foundation for mathematicians, logicians, statisticians, economists, architects, engineers, and many others to build and test their theories on.

證明提供一個穩固的基礎給數學家、邏輯學家、統計學家、 經濟學家、建築師、工程師、 還有許多其它人，讓他們得以在這基礎上建立並測試他們的理論。

And they're just plain awesome!

這簡直棒極了！

Let me start at the beginning.

讓我從頭說起。

I'll introduce you to a fellow named Euclid, as in, "Here's looking at you, Clid."

我將介紹一個人，他叫做歐基里得，就像是「就看你的了，小子」的那位。 （譯註：北非諜影臺詞，小子英文 kin 音似基里得）

He lived in Greece about 2,300 years ago, and he's considered by many to be the father of geometry.

他生活在約 2,300 年前的希臘，而大多數人認為他是幾何學之父。

So, if you've been wondering where to send your geometry fan mail, Euclid of Alexandria is the guy to thank for proofs.

所以如果你想該把幾何學粉絲郵件寄到哪裡的話，就證明學而言，亞歷山卓的歐基里得就是那位該感謝的人。

Euclid is not really known for inventing or discovering a lot of mathematics, but he revolutionized the way in which it is written, presented, and thought about.

歐基里得並不真的是以創造、發現大量數學而聞名，但是他改革了數學寫作、表述、及思考的方法。

Euclid set out to formalize mathematics by establishing the rules of the game.

歐基里得藉由訂定遊戲規則將數學公式化、條理化。

These rules of the game are called axioms.

這些規則被叫做公理。

Once you have the rules, Euclid says you have to use them to prove what you think is true.

只要有了規則，歐基里得說你必須用這些規則來證明你想的是對的。

If you can't, then your theorem or idea might be false.

如果你沒辦法做到，那麼你所想的定理就有可能是錯的。

And if your theorem is false, then any theorems that come after it and use it might be false, too.

而如果你的定理是錯的，那任何隨之而來的定理同樣也有可能是錯的。

Like how one misplaced beam can bring down the whole house.

就像是一個錯位的橫樑可以弄垮整棟房子一樣。

So, that's all that proofs are:

所以整個證明的過程就是：

Using well-established rules to prove beyond a doubt that some theorem is true.

利用完善的規則，合理地證明某些定理是正確的。

Then, you use those theorems like blocks to build mathematics.

接著把定理當做積木般來建造數學。

Let's check out an example.

我們來看看一個例子。

Say I want to prove that these two triangles are the same size and shape.

假設我們想要證明這兩個三角形大小一樣、形狀也一樣。

In other words, they are congruent.

換句話說，他們是全等的。

Well, one way to do that is to write a proof that shows that all three sides of one triangle are congruent to all three sides of the other triangle.

嗯，一個辦法是寫一段證明來說明一個三角形的三條邊和另一個三角形的三條邊分別都等長。

So, how do we prove it?

所以要怎麼做？

First, I'll write down what we know.

首先，我會寫下我們所知道的。

We know that point M is the midpoint of AB.

我們知道 M 點 是 AB 邊的中點。

We also know that sides AC and BC are already congruent.

我們也知道 AC 邊和 BC 邊本來就等長。

Now, let's see, what does the midpoint tell us?

現在咱們看看。這個中點可以告訴我們什麼？

Luckily, I know the definition of midpoint.

幸運地，我知道中點的定義。

It is basically the point in the middle.

基本上它就是正中央的那點。

What this means is that AM and BM are the same length, since M is the exact middle of AB.

它的意思就是 AM 邊和 BM 邊的長度相同，因為 M 點位在 AB 邊的正中間。

In other words, the bottom side of each of our triangles are congruent.

也就是說，我們考慮的三角形的兩個底邊是等長的。

I'll put that as step two.

我會把這當做第二步。

Great! So far, I have two pairs of sides that are congruent.

太棒了！目前為止我已經有兩組邊等長。

The last one is easy.

最後一步是簡單的。

The third side of the left triangle is CM, and the third side of the right triangle is, well, also CM.

左邊三角形的第三邊是 CM 邊，而右邊三角形的第三邊是……對，也是 CM 邊。

They share the same side; of course it's congruent to itself!

他們共用這條邊，當然和自己等長！

This is called the reflexive property⏤everything is congruent to itself.

這個叫做「反身性」，就是說每條邊都和自己等長。

I'll put this as step three.

我把這當做第三步。

Ta-da! You've just proven that all three sides of the left triangle are congruent to all three sides of the right triangle.

噠啦！你已經證明左邊三角形的三條邊和右邊三角形的完全等長。

Plus, the two triangles are congruent because of the side-side-side congruence theorem for triangles.

附帶地，兩個三角形會全等是由於三角形 SSS 全等性質。

When finished with a proof, I like to do what Euclid did.

當完成了一段證明，我喜歡做件歐基里得會做的事。

He marked the end of a proof with the letters QED.

他用字母 QED 來標記一段證明的結尾。

It's Latin for "quod erat demonstrandum", which translates literally to "what was to be proven".

這表示拉丁話「quod erat demonstrandum」，字面上的意思就是「這就是所要證明的」。

But I just think of it as "look what I just did!"

但是我只把它想成是：瞧瞧我做了什麼！

I can hear what you're thinking,

我可以聽見你正在想什麼：

"Why should I study proofs?"

「為什麼我要學證明？」

One reason is that they could allow you to win any argument.

一個理由是證明可以讓你在爭論中獲勝。

Abraham Lincoln, one of our nation's greatest leaders of all time used to keep a copy of Euclid's "Elements" on his bedside table to keep his mind in shape.

亞伯拉罕．林肯，一位美國整個時期最偉大的領導者習慣放一本歐基里得的《幾何原本》在他的桌邊來讓他的思考有條理。

Another reason is, you can make a million dollars.

另一個理由是你可以賺到一百萬美元。

You heard me, one million dollars.

你沒聽錯，一百萬美元。

That's the price that the Clay Mathematics Institute in Massachusetts is willing to pay anyone who proves one of the many unproven theories that it calls "the millennium problems".

這是麻州克雷數學研究所所提出的價格，將付給任何解出某些未知猜想的人，這些猜想被稱作「千禧年大獎難題」。

A couple of these have been solved in the '90s and 2000s.

其中有一些已經在 90 及 2000 年代被解決了。

But beyond money and arguments, proofs are everywhere.

但是超乎錢金和爭論的是，證明無所不在。

They underly architecture, art, computer programming, and internet security.

它們潛藏在建築、藝術、程式設計、以及網路安全之中。

If no one understood or could generate a proof, we could not advance these essential parts of our world.

如果都沒人了解、或是有辦法證明，我們將無法在這些重要領域中進步。

Finally, we all know that the proof is in the pudding.

最後，我們都知道證明藏在布丁裡。（譯註：美國諺語，表示布丁好吃的證明要吃了才知道。）

And pudding is delicious. QED.

而布丁是美味的，QED。