in the 1950s,

當尼古拉·布爾巴基 在 1950 年代

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

申請加入美國數學學會時，

He'd published articles in international journals

他已經是當時最有 影響力的數學家之一了。

and his textbooks were required reading.

他在國際期刊中發表文章，

Yet his application was firmly rejected for one simple reason—

他的教科書也被列為必讀。

Nicolas Bourbaki did not exist.

但他的申請卻被堅決駁回， 理由很簡單——

Two decades earlier, mathematics was in disarray.

尼古拉·布爾巴基並不存在。

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

二十年前，數學界一片混亂。

and the field had become fragmented.

許多知名的數學家 死於第一次世界大戰，

Different branches used disparate methodology to pursue their own goals.

讓這個領域變得支離破碎。

And the lack of a shared mathematical language

不同的分支用迥然不同的方法論 來追求他們自己的目標。

made it difficult to share or expand their work.

因為缺乏共同的數學語言，

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

他們很難分享或擴展研究。

While studying at the prestigious École normale supérieure,

1934 年，一群法國數學家 特別感到忍無可忍了。

they found the textbook for their calculus class so disjointed

在有聲望的巴黎 高等師範學院讀書時，

that they decided to write a better one.

他們發現微積分課的 教科書實在太沒條理了，

The small group quickly took on new members,

因此他們決定寫一本更好的。

and as the project grew, so did their ambition.

這個小組很快就招收了新成員，

The result was the "Éléments de mathématique,"

隨著計劃越做越大， 他們的野心也越來越大。

a treatise that sought to create a consistent logical framework

成果就是《數學原本》，

unifying every branch of mathematics.

這本專著旨在創造一致的邏輯框架，

The text began with a set of simple axioms—

統一數學的所有分支。

laws and assumptions it would use to build its argument.

正文的一開始就是 一組簡單的公理——

From there, its authors derived more and more complex theorems

該書用來建構論點的定律和假設。

that corresponded with work being done across the field.

該書的作者依此導出了 更多複雜的定理，

But to truly reveal common ground,

並對應到數學領域中的各種研究。

the group needed to identify consistent rules

但，若要真正揭示出共同點，

that applied to a wide range of problems.

這個小組就要找出一致的規則

To accomplish this, they gave new, clear definitions

套用在各種問題上。

to some of the most important mathematical objects,

為了完成這個目標，他們 針對一些最重要的數學物件

including the function.

做出新的、清楚的定義，

It's reasonable to think of functions as machines

包括「函數」。

that accept inputs and produce an output.

把函數看成機器是很合理的，

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

它能接受輸入，產生輸出。

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

但如果我們把函數視為 兩個群組之間的橋樑，

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

我們就可以開始主張 它們之間有某種邏輯關係。

We could define a function where every numerical input corresponds

比如，有一個數字群組 和一個字母群組。

to the same alphabetical output,

我們可以定義出一個函數， 讓每個輸入的數字

but this doesn't establish a particularly interesting relationship.

都對應到同樣的輸出字母，

Alternatively, we could define a function where every numerical input

但這樣做並不會建立出 特別有趣的關係。

corresponds to a different alphabetical output.

另一個選擇是定義一個函數，

This second function sets up a logical relationship

讓每個輸入數字都對應到 不同的輸出字母。

where performing a process on the input has corresponding effects

這第二個函數就設立了 一種邏輯關係，

on its mapped output.

針對輸入值進行一個流程，

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

就會讓對應的輸出值 產生相應的效應。

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

這個小組開始這樣定義函數：

they defined it as injective.

它如何對應不同定義域的元素。

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

如果函數的輸出值 來自唯一的輸入值，

the function was surjective.

他們就把它定義為單射函數。

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

如果每個輸出值都可以 對應到至少一個輸入值，

This allowed mathematicians to establish logic that could be translated

這個函數就是蓋射函數。

across the function's domains in both directions.

在對射函數中，

Their systematic approach to abstract principles

每個元素都剛好有一對一的對應值。

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

這麼一來，數學家所建立的邏輯

and an over-dependence on logic constrained creativity.

就能在該函教的定義域中 做雙向的轉換。

But this rebellious band of scholars gleefully ignored conventional wisdom.

他們針對抽象原則所做的系統化方法

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

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，

with their biggest stunt yet.

且過度仰賴由邏輯限制的創意。

They decided to publish "Éléments de mathématique"

但這群叛逆的學者很愉快地 無視了傳統的智慧。

and all their subsequent work under a collective pseudonym:

他們想要改革這個領域，

Nicolas Bourbaki.

並用他們最大的噱頭 來紀念這個時刻。

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

他們決定要出版《數學原本》

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

以及他們後續用共同的筆名

Their invented mathematician claimed to be a reclusive Russian genius

「尼古拉·布爾巴基」所做的研究。

who would only meet with his selected collaborators.

在接下來二十年，布爾巴基的 出版品成了標準的參考資料。

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

而小組的成員對這個 惡作劇的認真程度不亞於研究。

and publicly insulted anyone who doubted his existence.

他們宣稱這位捏造的數學家 是位隱居的俄國天才，

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

只會和他選中的合作夥伴見面。

the group ended their joke the only way they could.

他們以布爾巴基的名義發電報， 宣佈他女兒的婚禮，

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

且任何懷疑他是否存在的人 都會被公開辱罵。

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

1968 年，他們的計謀 無法再繼續下去了，

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

他們以唯一能用的 方式來結束這個玩笑。

Bourbaki's influence informs much current research.

他們印出布爾巴基的訃告， 文中盡是數學雙關語。

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

儘管布爾巴基表面上已故，

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

使用這個名字的數學家 至今仍然活著。