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Inside the secret math society known simply as Nicolas Bourbaki

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https://www.quantamagazine.org/inside-th...-20201109/

EXERPTS: For almost a century, the anonymous members of Nicolas Bourbaki have written books intended as pure expressions of mathematical thought. [...] Bourbaki began in 1934, the initiative of a small number of recent ENS alumni. Many of them were among the best mathematicians of their generation. But as they surveyed their field, they saw a problem. The exact nature of that problem is also the subject of myth.

In one telling, Bourbaki was a response to the loss of a generation of mathematicians to World War I [...] In a more prosaic but probably also more likely rendering, the original Bourbaki members were simply dissatisfied with the field’s textbooks and wanted to create better ones. “I think at the beginning it was just for that very concrete matter,” Chambert-Loir said.

Whatever their motivation, the founders of Bourbaki began to write. Yet instead of writing textbooks, they ended up creating something completely novel: free-standing books that explained advanced mathematics without reference to any outside sources. [...] the project quickly expanded, since it’s hard to explain one mathematical idea without involving many others. “They realized that if they wanted to do this cleanly, they needed [ideas from other areas], and Bourbaki grew and grew into something huge,” Gouëzel said.

The most distinctive feature of Bourbaki was the writing style: rigorous, formal and stripped to the logical studs. The books spelled out mathematical theorems from the ground up without skipping any steps — exhibiting an unusual degree of thoroughness among mathematicians. “In Bourbaki, essentially, there are no gaps,” Gouëzel said. “They are super precise.”

But that precision comes at a cost: Bourbaki books can be hard to read. They don’t offer a contextualizing narrative that explains where concepts come from, instead letting the ideas speak for themselves. “Essentially, you give no comment about what you do or why you do it,” Chambert-Loir said. “You state stuff and prove it, and that’s it.”

[...] focus on collaboration is also where the group’s insistence on anonymity comes from. ... “It’s sort of hard to imagine a group of young academics right now, people without permanent lifelong positions, devoting a huge amount of time to something they’ll never get credit for,” said Lillian Pierce of Duke University. “This group took this on in a sort of selfless way.”

Bourbaki quickly had an impact on mathematics. Some of the first books, published in the 1940s and ’50s, invented vocabulary that is now standard — terms like “injective,” “surjective” and “bijective,” which are used to describe properties of a map between two sets.

This was the first of two main periods in which Bourbaki was especially influential. The second came in the 1970s when the group published a series of books on Lie groups and Lie algebras that is “unanimously considered a masterpiece,” Chambert-Loir said.

Today, the influence of the group’s books has waned... (MORE - details)
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