Does math have a major flaw? - Jacqueline Doan and Alex Kazachek

翻译人员: J Zhong 校对人员: Yanyan Hong

Consider this mathematician,

这是一位数学家。

with her standard-issue infinitely sharp knife and a perfect ball.

她有一把绝对锋利的标准刀 和一个完美球体。

She frantically slices and distributes the ball into an infinite number of boxes.

她手起刀落将球切分， 并把切片分别装入无数个盒子中，

She then recombines the parts into five precise sections.

然后将这些切片 分成完全平均的五份。

Gently moving and rotating these sections around,

通过仔细地移动和旋转这些切片，

seemingly impossibly, she recombines them to form two identical, flawless,

她竟然能将这些切片 重组为两个完整无缝的球体，

and complete copies of the original ball.

且与原本的球体完全一致。

This is a result known in mathematics as the Banach-Tarski paradox.

这个结果在数学中 被称为巴纳赫-塔尔斯基悖论。

The paradox here is not in the logic or the proof—

这里的悖论不是 因逻辑或证据产生的——

which are, like the balls, flawless—

毕竟它们就像 那些球一样完美无缺——

but instead in the tension between mathematics

而在于数学 与我们现实体验之间的矛盾。

and our own experience of reality.

And in this tension lives some beautiful and fundamental truths

于此矛盾之中，我们能够窥见 一些迷人的基本真理，

about what mathematics actually is.

从而理解数学的本质。

We’ll come back to that in a moment, but first,

这个问题先放在一边。 首先，

we need to examine the foundation of every mathematical system: axioms.

我们需要检视一下 所有数学系统的基础：公理。

Every mathematical system is built and advanced

每个数学系统都是通过

by using logic to reach new conclusions.

使用逻辑得出新结论来构建和推进的。

But logic can’t be applied to nothing;

但是逻辑也需要应用的对象；

we have to start with some basic statements, called axioms,

我们必须从一些 被称为“公理”的基本陈述开始。

that we declare to be true, and make deductions from there.

我们声明这些陈述是正确的， 然后基于其进行推理。

Often these match our intuition for how the world works—

公理一般与我们的直觉 对世界的认知是相符的——

for instance, that adding zero to a number has no effect is an axiom.

如 “ 0 与数字相加不改变结果” 是一个公理。

If the goal of mathematics is to build a house, axioms form its foundation—

如果数学的目标是盖房子， 公理就是地基——

the first thing that’s laid down, that supports everything else.

地基是最早打下的， 支撑其他所有结构。

Where things get interesting is that by laying a slightly different foundation,

有趣的是， 通过打一个略有不同的地基，

you can get a vastly different but equally sound structure.

能够得到一个截然不同 但同样坚实的结构。

For example, when Euclid laid his foundations for geometry,

例如，欧几里得 为几何学奠定基础时，

one of his axioms implied that given a line and a point off the line,

由他其中一条公理可推理出： 给定一条直线和一个离线点，

only one parallel line exists going through that point.

只有一条平行线穿过该点。

But later mathematicians,

但是后来的数学家想知道

wanting to see if geometry was still possible without this axiom,

几何学没有这条公理是否还能成立，

produced spherical and hyperbolic geometry.

于是有了球面几何和双曲几何。

Each valid, logically sound, and useful in different contexts.

它们都有效且符合逻辑， 在不同的情况下非常有用。

One axiom common in modern mathematics is the Axiom of Choice.

现代数学中常用的一个公理 是选择公理。

It typically comes into play in proofs that require choosing elements from sets—

需要从集合中选择元素的证明中 通常会用到它——

which we’ll grossly simplify to marbles in boxes.

这种证明简单来说， 就好比盒装弹珠。

For our choices to be valid, they need to be consistent,

为了使我们的选择有效， 它们必须保持一致，

meaning if we approach a box, choose a marble,

意味着若从其中一个盒子中 选择一颗弹珠，

and then go back in time and choose again, we'd know how to find the same marble.

然后回到过去、再次选择， 我们就会知道如何找到同一颗弹珠。

If we have a finite number of boxes, that’s easy.

如果我们的盒子数量有限， 这便容易做到。

It’s even straightforward when there are infinite boxes

就算有无限个盒子也是小菜一碟，

if each contains a marble that’s readily distinguishable from the others.

只要盒中含有一颗 与其他弹珠不同的弹珠。

It’s when there are infinite boxes with indistinguishable marbles

然而，当有无限个盒子， 且装的全是无法区分的弹珠时，

that we have trouble.

就会非常难办。

But in these scenarios,

但是，在这种情况下，

the Axiom of Choice lets us summon a mysterious omniscient chooser

选择公理就好比召唤了 一个无所不知的神秘选择者，

that will always select the same marbles—

总是能选出相同的弹珠，

without us having to know anything about how those choices are made.

而无需我们知道 这些选择是如何做出的。

Our stab-happy mathematician, following Banach and Tarski’s proof,

我们的疯狂数学家 效仿巴纳赫和塔尔斯基的证明，

reaches a step in constructing the five sections

走到了将五个部分重组的那一步，

where she has infinitely many boxes filled with indistinguishable parts.

她已在无限多的盒子中 装入了无法区分的小切片。

So she needs the Axiom of Choice to make their construction possible.

因此，她需要选择公理 才有可能将它们复原。

If the Axiom of Choice can lead to such a counterintuitive result,

如果选择公理带来的结果 如此地反直觉，

should we just reject it?

我们不应该否认它吗？

Mathematicians today say no,

今天的数学家不这么认为，

because it’s load-bearing for a lot of important results in mathematics.

因为它是数学中 许多重要成果的根基。

Fields like measure theory and functional analysis,

测度论和泛函分析等领域

which are crucial for statistics and physics,

对于统计学和物理学至关重要；

are built upon the Axiom of Choice.

而它们都是建立在选择公理上的。

While it leads to some impractical results,

虽然选择公理能推理出 不切实际的结果，

it also leads to extremely practical ones.

但也能得到极其实用的结果。

Fortunately, just as Euclidean geometry exists alongside hyperbolic geometry,

幸运的是，就像欧几里得几何 与双曲几何共存一样，

mathematics with the Axiom of Choice coexists with mathematics without it.

使用选择公理和不使用它的数学 也是共存的。

The question for many mathematicians isn’t whether the Axiom of Choice,

对于许多数学家来说， 他们关注的并非这些公理的对错，

or for that matter any given axiom, is right or not,

不论是选择公理还是其他公理，

but whether it’s right for what you’re trying to do.

重要的是它们是否能 为你的目标服务。

The fate of the Banach-Tarski paradox lies in this choice.

巴纳赫-塔斯基悖论的命运 正由这种选择决定。

This is the freedom mathematics gives us.

这便是数学赋予我们的自由。

Not only is it a way to model our physical universe

这不仅是使用我们在 日常经历中由直觉得到的公理

using the axioms we intuit from our daily experiences,

模拟我们的物理宇宙的一种方式，

but a way to venture into abstract mathematical universes

我们还能得以漫游抽象数学宇宙，

and explore arcane geometries and laws unlike anything we can ever experience.

探索神秘的几何和定律， 这样的体验是绝无仅有的。

If we ever meet aliens, axioms which seem absurd and incomprehensible to us

若能遇见外星人， 这些貌似荒谬和难以理解的公理

might be everyday common sense to them.

对他们而言可能是日常常识。

To investigate, we might start by handing them an infinitely sharp knife

若想知道，我们可以交 给他们一把无限锋利的刀

and a perfect ball,

和一个完美的球，

and see what they do.

然后拭目以待。