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

譯者: Lilian Chiu 審譯者: SF Huang

Consider this mathematician,

想想這位數學家，

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

她有一把無限鋒利的標配刀 和一顆完美的球。

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

她瘋狂地切割這顆球，並將 切片分配到無限個盒子中。

She then recombines the parts into five precise sections.

然後，她將這些切片 重組成五個精確的部分。

Gently moving and rotating these sections around,

她輕輕地移動和旋轉這些部分，

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

看似不可能的是，

她將它們重新組合， 形成了原始那顆球的

and complete copies of the original ball.

兩個副製品，完全相同且毫無瑕疵。

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

在數學領域，這個結果 稱為巴拿赫—塔斯基悖論。

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

矛盾之處並不在於邏輯或證明——

which are, like the balls, flawless—

這些都和球一樣無瑕——

but instead in the tension between mathematics

反而是在於數學和我們自己

and our own experience of reality.

對現實的體驗之間的緊張關係。

And in this tension lives some beautiful and fundamental truths

在這種緊張關係中， 有著一些美麗且基本的真理，

about what mathematics actually is.

說明數學到底是什麼。

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

我們稍後再來談這一點，

但首先，我們要檢視 每個數學系統的基礎：

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

公理。

Every mathematical system is built and advanced

每個數學系統的建立和進展

by using logic to reach new conclusions.

都仰賴使用邏輯來得出新的結論。

But logic can’t be applied to nothing;

但什麼都沒有時就不能應用邏輯；

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

我們必須從一些基本陳述 來著手，稱為公理，

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

我們宣稱那些公理為真， 從那裡開始做推論。

Often these match our intuition for how the world works—

通常，這些公理會符合我們 直覺上世界運作的方式——

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

例如把一個數字加上零不會 有任何效果，這就是公理。

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

如果數學的目標是建造一間房子，

公理是用來形成它的地基——

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

是最先建造的部分， 用以支撐所有其他部分。

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

有趣的是，若建立略有不同的基礎，

you can get a vastly different but equally sound structure.

可以建出非常不同 但同樣健全的建築。

For example, when Euclid laid his foundations for geometry,

例如，當歐幾里德 為幾何學建立基礎時，

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

他的其中一個公理就暗示：給定 一條直線和一個不在線上的點，

only one parallel line exists going through that point.

只會有一條平行線通過該點。

But later mathematicians,

但後來的數學家，

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

因為想看看沒有這個公理時 幾何學是否仍有可能，

produced spherical and hyperbolic geometry.

便創造了球面幾何學和雙曲幾何學。

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

兩者都有根據，在邏輯上很合理，

且能用在不同情境中。

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

現代數學中常見的公理 之一就是選擇公理。

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

會用到它的情況通常是

在做證明時需要從集合中 選擇元素的情況，

which we’ll grossly simplify to marbles in boxes.

我們可以把這種情況大略 簡化為盒子中的大理石。

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

要讓我們的選擇是有效的， 這些大理石必須要一致，

meaning if we approach a box, choose a marble,

意即，當我們從一個盒子裡 選擇了一顆大理石，

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

接著我們回去再選擇一次，

我們還是會知道要如何 找到同樣的大理石。

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

如果盒子的數量有限，那就很容易。

It’s even straightforward when there are infinite boxes

這種情況也很直觀： 盒子的數量無限，

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

每個盒子內有一個可明顯 和其他大理石區別出來的大理石。

It’s when there are infinite boxes with indistinguishable marbles

但當盒子的數量無限， 大理石又無法區別時，

that we have trouble.

就麻煩了。

But in these scenarios,

但在這些情境中，選擇公理

the Axiom of Choice lets us summon a mysterious omniscient chooser

讓我們召喚一名神秘的全知選擇者，

that will always select the same marbles—

他永遠會選到同樣的大理石——

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

且不需要我們知道 這些選擇是如何做出來的。

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

我們這位用刀的快樂數學家

遵循巴拿赫—塔斯基的證明，

reaches a step in constructing the five sections

來到了這個步驟： 要建造出五個部分，

where she has infinitely many boxes filled with indistinguishable parts.

此時她有無限個盒子， 內有無法區別的切片。

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

因此，她需要選擇公理， 才有可能建造出這五個部分。

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

如果選擇公理可以帶出 如此反直覺的結果，

should we just reject it?

我們該捨棄它嗎？

Mathematicians today say no,

今天的數學家說不行，

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

因為許多重要的數學結果 都立基在這個公理上。

Fields like measure theory and functional analysis,

測度論和泛函分析等領域

which are crucial for statistics and physics,

對於統計學和物理學都至關重要，

are built upon the Axiom of Choice.

它們都立基在選擇公理上。

While it leads to some impractical results,

雖然此公理會導出 一些不實際的結果，

it also leads to extremely practical ones.

但也會導出非常實用的結果。

Fortunately, just as Euclidean geometry exists alongside hyperbolic geometry,

幸運的是，正如歐幾里德 幾何學與雙曲幾何學並存一樣，

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

有選擇公理的數學

也與沒有選擇公理的數學並存。

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

對於許多數學家來說，

問題並不在於選擇公理

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

或任何給定的公理是否正確，

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

而是針對你想達到的 目的，它是否正確。

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

巴拿赫—塔斯基悖論的命運 取決於這個選擇。

This is the freedom mathematics gives us.

這是數學給我們的自由。

Not only is it a way to model our physical universe

它不僅是一種利用我們 從日常經驗中直覺知道的公理

using the axioms we intuit from our daily experiences,

來形塑實體宇宙的方法，

but a way to venture into abstract mathematical universes

還是一種探索抽象數學領域的方法，

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

探究我們不曾經歷過的 神秘幾何學和定律。

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

如果我們遇見外星人，

在我們眼中很荒唐且難以理解的公理

might be everyday common sense to them.

對他們來說可能是日常的常識。

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

如果想探究，我們首先可以 交給他們一把無限鋒利的刀

and a perfect ball,

和一顆完美的球，

and see what they do.

看看他們會怎麼做。