The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy

譯者: 盈蓓 余 審譯者: Helen Chang

Consider the following sentence: “This statement is false.”

請看這個句子： 「這個表述是假的。」

Is that true?

是真的嗎？

If so, that would make this statement false.

如果是的話 這個表述就會變成假的

But if it’s false, then the statement is true.

但如果它是假的 那這個表述就是真的

By referring to itself directly, this statement creates an unresolvable paradox.

透過直接說明句子本身 它製造出無法解決的悖論

So if it’s not true and it’s not false— what is it?

所以如果它既不真又不假 它是什麼？

This question might seem like a silly thought experiment.

這個問題可能聽起來像 一個無厘頭的思考實驗

But in the early 20th century, it led Austrian logician Kurt Gödel

但它在二十世紀初 讓澳洲邏輯學家

to a discovery that would change mathematics forever.

庫爾特．哥德爾 發現永遠改變數學的東西

Gödel’s discovery had to do with the limitations of mathematical proofs.

哥德爾的發現 跟數學證明的限制有關

A proof is a logical argument that demonstrates

證明是邏輯論證，用來展示

why a statement about numbers is true.

何以某個跟數字相關的表述為真

The building blocks of these arguments are called axioms—

這些論證的構成要素 被稱為公理

undeniable statements about the numbers involved.

關於其中數字 不證自明的論述

Every system built on mathematics,

每個以數學為基礎的系統

from the most complex proof to basic arithmetic,

從最複雜的證明到基本的算數

is constructed from axioms.

都是由公理構成的

And if a statement about numbers is true,

若一個關於數字的表述為真

mathematicians should be able to confirm it with an axiomatic proof.

數學家理應能夠 運用公理化的證明證實它

Since ancient Greece, mathematicians used this system

從古希臘時代開始 數學家們運用這個系統

to prove or disprove mathematical claims with total certainty.

以完全的確定性 證明或推翻數學主張

But when Gödel entered the field,

但當哥德爾進入這個領域時

some newly uncovered logical paradoxes were threatening that certainty.

某些新發現的邏輯悖論 正在威脅那股確定性

Prominent mathematicians were eager to prove

著名數學家們渴望證明 數學沒有任何矛盾

that mathematics had no contradictions.

Gödel himself wasn’t so sure.

哥德爾本人卻不那麼確定

And he was even less confident that mathematics was the right tool

甚至還沒把握數學是不是

to investigate this problem.

研究這個問題的正確工具

While it’s relatively easy to create a self-referential paradox with words,

雖然運用文字創造 自我指涉的悖論比較簡單

numbers don't typically talk about themselves.

但數字通常不會指涉自身

A mathematical statement is simply true or false.

一個數學相關的表述 不是真的就是假的

But Gödel had an idea.

但哥德爾有個想法

First, he translated mathematical statements and equations into code numbers

首先，他把數學相關的表述 和方程式轉譯成代碼

so that a complex mathematical idea could be expressed in a single number.

如此，一個複雜的數學概念 就可以被用單一數字表述

This meant that mathematical statements written with those numbers

這表示運用那些數字寫出的 數學相關表述

were also expressing something about the encoded statements of mathematics.

同時在表示 被編碼的數學表述相關的東西

In this way, the coding allowed mathematics to talk about itself.

用這樣的方式 編碼讓數學能夠自述

Through this method, he was able to write:

透過這個方法 他可以用方程式寫出：

“This statement cannot be proved” as an equation,

「這個表述不能被證實」

creating the first self-referential mathematical statement.

他以此創造了第一個 自我指涉悖論的數學表述

However, unlike the ambiguous sentence that inspired him,

然而，不同於那些 帶給他靈感的歧異句

mathematical statements must be true or false.

數學表述必須要是真或假

So which is it?

所以是哪個呢？

If it’s false, that means the statement does have a proof.

如果它是假的 就表示這個表述被證明了

But if a mathematical statement has a proof, then it must be true.

但如果一個數學表述被證明了 它一定是真的

This contradiction means that Gödel’s statement can’t be false,

這個矛盾表示 哥德爾的表述不可能是假的

and therefore it must be true that “this statement cannot be proved.”

因此「這個表述不能被證明」 一定是真的

Yet this result is even more surprising,

所以結果更讓人驚訝

because it means we now have a true equation of mathematics

因為它表示我們現在有個 真的數學方程式

that asserts it cannot be proved.

主張自己無法被證實

This revelation is at the heart of Gödel’s Incompleteness Theorem,

這個真相是 哥德爾不完備定理的中心

which introduces an entirely new class of mathematical statement.

它引介出一個 數學表述的全新類別

In Gödel’s paradigm, statements still are either true or false,

在哥德爾的範例中 表述仍是要嘛真要嘛假

but true statements can either be provable or unprovable

但是真的表述 在一組既有的公理中

within a given set of axioms.

能夠是可證的或不可證的

Furthermore, Gödel argues these unprovable true statements

此外哥德爾認為 這些不可證的真實表述

exist in every axiomatic system.

存在每個公理系統當中

This makes it impossible to create

這讓用數學創造 一個完美且完整的系統

a perfectly complete system using mathematics,

變得不可能

because there will always be true statements we cannot prove.

因為永遠會有 我們無法證明的真實表述

Even if you account for these unprovable statements

即使你將這些不可證表述 以新公理的身份

by adding them as new axioms to an enlarged mathematical system,

加入一個更大的數學系統 來說明它們

that very process introduces new unprovably true statements.

這個過程又會引介出新的 不可證的真表述

No matter how many axioms you add,

不管你加入多少公理

there will always be unprovably true statements in your system.

永遠都會有不可證的真表述 存在你的系統當中

It’s Gödels all the way down!

這就是哥德爾的一路向前

This revelation rocked the foundations of the field,

這個發現 動搖了數學領域的基礎

crushing those who dreamed that every mathematical claim would one day

震驚那些夢想著總有一天 所有數學主張

be proven or disproven.

都將被證實或推翻的人

While most mathematicians accepted this new reality, some fervently debated it.

雖然大部分數學家 接受了這個嶄新的現實

有些人展開激烈的爭論 而其他人則仍然嘗試忽略

Others still tried to ignore the newly uncovered a hole

將這個新發現視為 領域中心的破洞

in the heart of their field.

But as more classical problems were proven to be unprovably true,

但隨著越來越多的經典問題 被證實為不可證之真

some began to worry their life's work would be impossible to complete.

有些人開始擔心 他們畢生的努力不可能完成

Still, Gödel’s theorem opened as many doors as a closed.

儘管如此，哥德爾的理論 敞開了許多大門

Knowledge of unprovably true statements

關於不可證的真表述的知識

inspired key innovations in early computers.

啟發了早期電腦的關鍵發展

And today, some mathematicians dedicate their careers

現今有些數學家的生涯致力於

to identifying provably unprovable statements.

發現能夠證明不可證的表述

So while mathematicians may have lost some certainty,

因此，雖然數學家可能 失去了一點確信

thanks to Gödel they can embrace the unknown

多虧了哥德爾 他們在任何探詢真理的核心

at the heart of any quest for truth.

都可以擁抱未知