The greatest mathematician that never lived - Pratik Aghor

譯者: Lilian Chiu 審譯者: Pui-Ching Siu

When Nicolas Bourbaki applied to the American Mathematical Society

當尼古拉·布爾巴基 在 1950 年代

申請加入美國數學學會時，

in the 1950s,

he was already one of the most influential mathematicians of his time.

他已經是當時最有 影響力的數學家之一了。

He’d published articles in international journals

他在國際期刊中發表文章，

and his textbooks were required reading.

他的教科書也被列為必讀。

Yet his application was firmly rejected for one simple reason—

但他的申請卻被堅決駁回， 理由很簡單——

Nicolas Bourbaki did not exist.

尼古拉·布爾巴基並不存在。

Two decades earlier, mathematics was in disarray.

二十年前，數學界一片混亂。

Many established mathematicians had lost their lives in the first World War,

許多知名的數學家 死於第一次世界大戰，

and the field had become fragmented.

讓這個領域變得支離破碎。

Different branches used disparate methodology to pursue their own goals.

不同的分支用迥然不同的方法論 來追求他們自己的目標。

And the lack of a shared mathematical language

因為缺乏共同的數學語言，

made it difficult to share or expand their work.

他們很難分享或擴展研究。

In 1934, a group of French mathematicians were particularly fed up.

1934 年，一群法國數學家 特別感到忍無可忍了。

While studying at the prestigious École normale supérieure,

在有聲望的巴黎 高等師範學院讀書時，

they found the textbook for their calculus class so disjointed

他們發現微積分課的 教科書實在太沒條理了，

that they decided to write a better one.

因此他們決定寫一本更好的。

The small group quickly took on new members,

這個小組很快就招收了新成員，

and as the project grew, so did their ambition.

隨著計劃越做越大， 他們的野心也越來越大。

The result was the "Éléments de mathématique,"

成果就是《數學原本》，

a treatise that sought to create a consistent logical framework

這本專著旨在創造一致的邏輯框架，

unifying every branch of mathematics.

統一數學的所有分支。

The text began with a set of simple axioms—

正文的一開始就是 一組簡單的公理——

laws and assumptions it would use to build its argument.

該書用來建構論點的定律和假設。

From there, its authors derived more and more complex theorems

該書的作者依此導出了 更多複雜的定理，

that corresponded with work being done across the field.

並對應到數學領域中的各種研究。

But to truly reveal common ground,

但，若要真正揭示出共同點，

the group needed to identify consistent rules

這個小組就要找出一致的規則

that applied to a wide range of problems.

套用在各種問題上。

To accomplish this, they gave new, clear definitions

為了完成這個目標，他們 針對一些最重要的數學物件

to some of the most important mathematical objects,

做出新的、清楚的定義，

including the function.

包括「函數」。

It’s reasonable to think of functions as machines

把函數看成機器是很合理的，

that accept inputs and produce an output.

它能接受輸入，產生輸出。

But if we think of functions as bridges between two groups,

但如果我們把函數視為 兩個群組之間的橋樑，

we can start to make claims about the logical relationships between them.

我們就可以開始主張 它們之間有某種邏輯關係。

For example, consider a group of numbers and a group of letters.

比如，有一個數字群組 和一個字母群組。

We could define a function where every numerical input corresponds

我們可以定義出一個函數， 讓每個輸入的數字

都對應到同樣的輸出字母，

to the same alphabetical output,

but this doesn’t establish a particularly interesting relationship.

但這樣做並不會建立出 特別有趣的關係。

Alternatively, we could define a function where every numerical input

另一個選擇是定義一個函數，

讓每個輸入數字都對應到 不同的輸出字母。

corresponds to a different alphabetical output.

This second function sets up a logical relationship

這第二個函數就設立了 一種邏輯關係，

where performing a process on the input has corresponding effects

針對輸入值進行一個流程，

就會讓對應的輸出值 產生相應的效應。

on its mapped output.

The group began to define functions by how they mapped elements across domains.

這個小組開始這樣定義函數：

它如何對應不同定義域的元素。

If a function’s output came from a unique input,

如果函數的輸出值 來自唯一的輸入值，

they defined it as injective.

他們就把它定義為單射函數。

If every output can be mapped onto at least one input,

如果每個輸出值都可以 對應到至少一個輸入值，

the function was surjective.

這個函數就是蓋射函數。

And in bijective functions, each element had perfect one to one correspondence.

在對射函數中，

每個元素都剛好有一對一的對應值。

This allowed mathematicians to establish logic that could be translated

這麼一來，數學家所建立的邏輯

就能在該函教的定義域中 做雙向的轉換。

across the function’s domains in both directions.

Their systematic approach to abstract principles

他們針對抽象原則所做的系統化方法

was in stark contrast to the popular belief that math was an intuitive science,

與一般的想法成鮮明對比： 一般的想法認為數學是直覺式科學，

and an over-dependence on logic constrained creativity.

且過度仰賴由邏輯限制的創意。

But this rebellious band of scholars gleefully ignored conventional wisdom.

但這群叛逆的學者很愉快地 無視了傳統的智慧。

They were revolutionizing the field, and they wanted to mark the occasion

他們想要改革這個領域，

並用他們最大的噱頭 來紀念這個時刻。

with their biggest stunt yet.

They decided to publish "Éléments de mathématique"

他們決定要出版《數學原本》

and all their subsequent work under a collective pseudonym:

以及他們後續用共同的筆名

「尼古拉·布爾巴基」所做的研究。

Nicolas Bourbaki.

Over the next two decades, Bourbaki’s publications became standard references.

在接下來二十年，布爾巴基的 出版品成了標準的參考資料。

And the group’s members took their prank as seriously as their work.

而小組的成員對這個 惡作劇的認真程度不亞於研究。

Their invented mathematician claimed to be a reclusive Russian genius

他們宣稱這位捏造的數學家 是位隱居的俄國天才，

who would only meet with his selected collaborators.

只會和他選中的合作夥伴見面。

They sent telegrams in Bourbaki’s name, announced his daughter’s wedding,

他們以布爾巴基的名義發電報， 宣佈他女兒的婚禮，

and publicly insulted anyone who doubted his existence.

且任何懷疑他是否存在的人 都會被公開辱罵。

In 1968, when they could no longer maintain the ruse,

1968 年，他們的計謀 無法再繼續下去了，

the group ended their joke the only way they could.

他們以唯一能用的 方式來結束這個玩笑。

They printed Bourbaki’s obituary, complete with mathematical puns.

他們印出布爾巴基的訃告， 文中盡是數學雙關語。

Despite his apparent death, the group bearing Bourbaki’s name lives on today.

儘管布爾巴基表面上已故，

使用這個名字的數學家 至今仍然活著。

Though he’s not associated with any single major discovery,

雖然布爾巴基沒有 完成任何重大發現，

Bourbaki’s influence informs much current research.

但近代的研究仍然受到他的影響。

And the modern emphasis on formal proofs owes a great deal to his rigorous methods.

且現代會強調正式證據的重要性，

很大部分都歸功於他的嚴格方法。

Nicolas Bourbaki may have been imaginary— but his legacy is very real.

雖然尼古拉·布爾巴基是個虛構人物，

他留下的貢獻卻非常真實。