An introduction to mathematical theorems - Scott Kennedy

翻译人员: Minji Seo 校对人员: Yolanda Zhang

What is proof?

什么是证明？

And why is it so important in mathematics?

为什么证明在数学中 是如此重要？

Proofs provide a solid foundation for mathematicians

证明提供了一个稳固的基础给数学家、

logicians, statisticians, economists, architects, engineers,

逻辑学家、统计学家、 经济学家、建筑师、工程师、

and many others to build and test their theories on.

还有许多其它人，让他们在这基础上 建立并测试他们的理论。

And they're just plain awesome!

这简直棒极了！

Let me start at the beginning.

让我从头说起。

I'll introduce you to a fellow named Euclid.

我将介绍一个人，他叫做欧基里得。

As in, "here's looking at you, Clid."

就像是“就看你的了，宝贝”的那位。 （北非谍影台词；“宝贝”英文音似“基里得”）

He lived in Greece about 2,300 years ago,

他生活在约 2300年前的希腊，

and he's considered by many to be the father of geometry.

而大多数人认为他是几何学之父。

So if you've been wondering where to send your geometry fan mail,

所以如果身为几何粉丝的你 在犹豫粉丝邮件要送到哪的话，

Euclid of Alexandria is the guy to thank for proofs.

对于证明来说，亚历山大的欧基里得 就是那位该感谢的人。

Euclid is not really known for inventing or discovering a lot of mathematics

欧基里得并不真的是以 创造、发现大量数学算法而闻名，

but he revolutionized the way in which it is written,

但是他改革了数学写作、

presented, and thought about.

表述以及思考的方法。

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

欧基里得借由订制定游戏规则 来将数学公式化、条理化。

These rules of the game are called axioms.

这些规则被叫做“定理”。

Once you have the rules,

只要有了规则，

Euclid says you have to use them to prove what you think is true.

欧基里得说你必须用这些规则 来证明你想的是对的。

If you can't, then your theorem or idea

如果你没办法做到，那么你所想的定理

might be false.

就有可能是错的。

而如果你的定理是错的， 那任何衍生出来的定理

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

might be false too.

同样也有可能是错的。

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

就像是一个错位的横梁 可以弄垮整栋房子一样。

So that's all that proofs are:

所以整个证明的过程就是：

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

利用完善的规则，合理地证明 某些定理是正确的。

Then you use those theorems like blocks

接着把定理当做积木，

to build mathematics.

来建造数学的大厦。

Let's check out an example.

我们来看看一个例子。

Say I want to prove that these two triangles

假设我们想要证明这两个三角形

are the same size and shape.

大小一样、形状也一样。

In other words, they are congruent.

换句话说，他们是全等的。

Well, one way to do that is to write a proof

那么，一个办法是写一段证明

that shows that all three sides of one triangle

来说明一个三角形的三条边

are congruent to all three sides of the other triangle.

和另一个三角形的三条边 分别都等长。

So how do we prove it?

那么要怎么做呢？

First, I'll write down what we know.

首先，我会写下所有的已知条件。

We know that point M is the midpoint of AB.

我们知道M点是AB边的中点。

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

我们也知道AC边和BC边本来就等长。

Now let's see. What does the midpoint tell us?

现在咱们看看， 这个中点可以告诉我们什么？

Luckily, I know the definition of midpoint.

很幸运，我知道中点的定义。

It is basically the point in the middle.

基本上它就是——正中央的那点！

What this means is that AM and BM are the same length,

它的意思就是AM边和BM边的长度相同，

since M is the exact middle of AB.

因为M点在AB边的正中间。

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

也就是说，我们考虑的三角形的 两个底边是等长的。

I'll put that as step two.

我会把这当做第二步。

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

太棒了！目前为止我已经有 两组边是等长的。

The last one is easy.

最后一步就简单了。

The third side of the left triangle

左边三角形的第三条边

is CM, and the third side of the right triangle is -

是CM边， 而右边三角形的第三边是......

well, also CM.

对，也是CM边。

They share the same side.

它们共享这条边。

Of course it's congruent to itself!

当然这条边和自己等长！

This is called the reflexive property.

这个叫做“反身性”。

Everything is congruent to itself.

就是说每条边都和自己等长。

I'll put this as step three.

我把这当做第三步。

Ta dah! You've just proven that all three sides of the left triangle

成功啦！ 你已经证明左边三角形的三条边

are congruent to all three sides of the right triangle.

和右边三角形的完全等长。

Plus, the two triangles are congruent

而且，两个三角形会全等，

because of the side-side-side congruence theorem for triangles.

是由于有三角形的“三边等长”定理。

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

当完成了一段证明，我喜欢做件 欧基里得会做的事。

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

他用字母 QED 来标记一段证明的结尾。

It's Latin for "quod erat demonstrandum,"

就是拉丁语中的 “quod erat demonstrandum”，

which translates literally to

字面上的意思就是

"what was to be proven."

“这就是所要证明的。”

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

但是我只把它想成是： “瞧瞧我做了什么！”

I can hear what you're thinking:

我可以听见你正在想什么：

why should I study proofs?

为什么我要学证明？

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

一个理由是，证明可以让你 在任何争论中获胜。

Abraham Lincoln, one of our nation's greatest leaders of all time

亚伯拉罕·林肯，美国历史上一位 最伟大的领导者，

used to keep a copy of Euclid's Elements on his bedside table

习惯放一本欧基里得的《几何原本》 在他的桌边，

to keep his mind in shape.

好让他的思考有条理。

Another reason is you can make a million dollars.

另一个理由是你可以赚到一百万美元。

You heard me.

你没听错。

一百万美元。

One million dollars.

这是麻州克雷数学研究所 所提出的价格，

That's the price that the Clay Mathematics Institute in Massachusetts

将付给任何解出 某些未证实的猜想的人，

is willing to pay anyone who proves one of the many unproven theories

that it calls "the millenium problems."

这些猜想被称作“千禧年大奖难题”。

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

其中有一些已经在 上世纪90年代和本世纪初被解决了。

But beyond money and arguments,

但是超乎金钱和争论的是，

proofs are everywhere.

证明无所不在。

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

它们潜藏在建筑、艺术、程序设计、 以及网络安全之中。

If no one understood or could generate a proof,

如果都没人了解、或是有办法证明，

we could not advance these essential parts of our world.

我们将无法在这些重要领域中进步。

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

最后，我们都知道证明藏在布丁里， （美国谚语：要证明布丁好吃，得吃了才知道）

And pudding is delicious. QED.

而布丁是美味的，QED。