For most of us, two degrees Celsius is a tiny difference in temperature, not even enough to make you crack a window.
对于我们大多数人来说，2摄氏度的温度变化并没有多大区别，还不足以让你开关窗户。
But scientists have warned that as CO2 levels in the atmosphere rise,
但是科学家警告由于大气中二氧化碳浓度的升高，
an increase in the Earth's temperature by even this amount can lead to catastrophic effects all over the world.
地球上即便升高这一点点温度，将会在全球导致一个灾难性的影响。
How can such a small measurable change in one factor lead to massive and unpredictable changes in other factors?
这样单个可测因素的微小变化是怎样导致其他因素发生大量不可预测的变化的呢？
The answer lies in the concept of a mathematical tipping point,
答案就是数学上的一个临界点的概念，
which we can understand through the familiar game of billiards.
我们可以从熟悉的桌球游戏中理解。
The basic rule of billiard motion is that a ball will go straight until it hits a wall,
桌球的基本运动规律就是它会沿着直线运动，直到撞到壁上，
then bounce off at an angle equal to its incoming angle.
会有一个与进入角度相等的反弹角。
For simplicity's sake, we'll assume that there is no friction, so balls can keep moving indefinitely.
为了简化，我们假设没有摩擦，所以球的运动就有很不确定性。
And to simplify the situation further, let's look at what happens with only one ball on a perfectly circular table.
为了使情况更加简单，让我们看看只让一个球在一个绝对圆的桌上运动会发生什么。
As the ball is struck and begins to move according to the rules, it follows a neat star-shaped pattern.
当球被击中并开始按规律运动的时候，会形成一个平滑的星形图案。
If we start the ball at different locations, or strike it at different angles,
如果我们在不同的位置或角度开球，
some details of the pattern change, but its overall form remains the same.
图案上的一些细节就会受到影响，但是整体情况是一样的。
With a few test runs, and some basic mathematical modeling,
经过一些测试和一些基本的数学模型分析后，
we can even predict a ball's path before it starts moving, simply based on its starting conditions.
仅仅简单依靠开球的条件，我们能在球运动之前就预测它的路线。
But what would happen if we made a minor change in the table's shape
但是如果我们对桌子的形状做一小点改变
by pulling it apart a bit, and inserting two small straight edges along the top and bottom?
我们将桌子分成两半，在顶部和底部插入两条直边，会发生什么呢？
We can see that as the ball bounces off the flat sides, it begins to move all over the table.
我们会看到，随着球从平整的边弹出，它开始在整个桌上运动。
The ball is still obeying the same rules of billiard motion,
它同样遵循着桌球的运动规律，
but the resulting movement no longer follows any recognizable pattern.
但是运动的结果就不再是我们认识的模型了。

With only a small change to the constraints under which the system operates,
在这个系统运行的过程中，我们只改变了一小点约束条件，
we have shifted the billiard motion from behaving in a stable and predictable fashion, to fluctuating wildly,
就改变了这个桌球的运动模式，从可以预测的、稳定的运动，到混乱的、不受控制的运动模式。
thus creating what mathematicians call chaotic motion.
因此创造了一个数学家数学家称作无序运动模型的东西。
Inserting the straight edges into the table acts as a tipping point,
在（圆形）桌子上放置两条直边作为临界点，
switching the systems behavior from one type of behavior, to another type of behavior.
将这个系统的规律运动变成了无序运动。
So what implications does this simple example have for the much more complicated reality of the Earth's climate?
对于复杂的地球气候问题，我们能从这个简单的例子中得到什么启发呢？
We can think of the shape of the table as being analogous to the CO2 level and Earth's average temperature:
我们可以把桌子的形状比作大气中二氧化碳的水平，或是地球的平均温度：
Constraints that impact the system's performance in the form of the ball's motion or the climate's behavior.
在这个桌球运动或说是气候变化的模型中，这两者是影响这个系统表现的条件。
During the past 10,000 years, the fairly constant CO2 atmospheric concentration of 270 parts per million
在过去的一万年里，二氧化碳在大气中的当量集中在百万分之二百七十左右，
kept the climate within a self-stabilizing pattern, fairly regular and hospitable to human life.
以此保证整体系统保持稳定，这对于人类的生活相当规律和适宜。
But with CO2 levels now at 400 parts per million,
但现在二氧化碳水平达到了百万分之四百左右的水平，
and predicted to rise to between 500 and 800 parts per million over the coming century, we may reach a tipping point
而且在下个世纪可能达到五百至八百，我们就可能到了“临界点”了。
where even a small additional change in the global average temperature
那时，在全球平均温度范围内一个小的附加变化，
would have the same effect as changing the shape of the table, leading to a dangerous shift in the climate's behavior,
可能与桌球实验中的变化有同样的影响，导致非常危险的气候变化，
with more extreme and intense weather events, less predictability, and most importantly, less hospitably to human life.
出现更多难以预测的极端的、激烈的天气事件，更重要的是，越发不适宜人类生活。
The hypothetical models that mathematicians study in detail may not always look like actual situations,
数学家们仔细研究的这些假想模型可能与我们的实际情况有所不同，
but they can provide a framework and a way of thinking that can be applied to help understand the more complex problems of the real world.
但是它提供给我们一个思考的框架和方法，能被用于理解更多真实生活中更复杂的问题。
In this case, understanding how slight changes in the constraints impacting a system can have massive impacts
在这种情况下，理解在系统中一个微小的条件变化的影响怎样导致一个更大的影响，
gives us a greater appreciation for predicting the dangers that we cannot immediately percieve with our own senses.
给我们带来了更多对目前凭我们的感官不能鉴别的预测危险的能力。
Because once the results do become visible, it may already be too late.
因为一旦我们看到这个结果，那时已经太迟了。