Now, in modern times, we call this Pythagoras' theorem.

在现代，我们称之为毕达哥拉斯定理。

Since these results cannot be boiled down to a mathematical theorem that we need to prove, we have to do it some other way.

因为这些结果不能被归结于一个我们需要去证明的数学定理，我们必须换种方式。

The legend around this chalk is that it's impossible to write a false theorem.

传说，用这款粉笔书写定理怎写都不会出错。

It's actually quite useful to have case studies or stories, which are these famous theorems.

实上，学习著名定理的案例和故当有用的。

The shape of the earth? The heliocentric universe? The Pythagorean theorem? The structure of DNA?

放弃对地球形状的认识？对以太阳为中心的宇宙的认识？还对勾股定理的认识？对DNA结构的认识？

Well, Kurt Godel's incompleteness theorems eliminate the possibility of a unified theory.

数学家库尔特·哥德尔的不完备定理排除了统一理论的可能性。

You might remember the Pythagorean theorem from school like what the formula is?

你可能还记得在学校学的勾股定理，比如公式？

And we did that in kind of the intuition behind the binomial theorem videos.

我们在利用二项式定理的方法来解决问题。

And yet, somehow, he came up with these incredible theorems.

然而，不知何故，他提出了那些令人难以置信的定理。

The third in our series looks at the Stolper-Samuelson theorem.

这一系列的第三个我们来看看斯托尔帕-萨缪尔森定理。

Weitzmann himself applied such an approach to climate change, formulating his " dismal theorem" , which states that, in the presence of sufficiently big risks with a small chance of great harm, regular cost-benefit analysis is of little use.

魏茨曼本人将这种方法应用于气候变化，并阐述了他的“悲观定理”，该定理指出，在存在概率小但伤害大的高风险情况下，常规的成本效益分析几乎没有用处。

And how. I once created a campaign set in 1940s London where Alan Turing was a character, and I had the mathematicians be magic users with theorems instead of spells.

当然啦，我曾经设计过一场战役背景为 1940 年代的伦敦，阿兰·图灵游戏角色，我让数学家们当法师，他们的魔法定理而不念咒。

According to the converse of the Pythagorean theorem, that has to make a right triangle, and, therefore, a square corner.

根据毕达哥拉斯的逆定理，这就可以形成一个直角三角形，因此，便可形成直角。

To prove the theorem we shall suppose that the graph G is drawn on a sphere as described above.

为了证明这个定理，我们假定这个圆G能按上述方式画在一个球面上。

He showed that if Fermat's last theorem were false, certain elliptical curves would have to exist.

他证明了，如果费马大定理为假，那某些椭圆曲线必然存在。

They all came up with elegant proofs for the famous Pythagorean theorem; the rule that says for a right triangle, the square of one side plus the square of the other side is equal to the square of the hypotenuse.

他们都对毕达哥拉斯定理（勾股定理）做出了精彩的证明；这个定理说，对于一个直角三角形，一边的平方加上另一边的平方等于斜边的平方。

Now they didn't have what we call the theorem.

那时他们还没有我们所说的定理。

Start with basic counting and work your way up to the many types of infinity culminating in the profundity of Cantor's theorem.

从基本的计数开始，一直到许多类型的无穷大，最终达到康托尔定理的深奥程度。

Instead, they knew all these particular cases where the theorem held true.

反，他们知道所有这些定理成立的特殊情况。

Proofs use existing mathematical rules and logic to demonstrate that a theorem must hold true all the time.

利用现有的数学定理和逻辑，我们可以证明该定理总成立。