Inside the Secret Math Society Known Simply as Nicolas Bourbaki

antoine chambert-loir’s initiation into one of math’s oldest secret societies began witha phone call. “they told me bourbaki ‘d like me to come n'see if i’d work w'dem,” he said.

chambert-loir accepted, and for a week in sep 2001 he spent 7 hrs a dy reading math texts out loud and discussing them w'da members of the group, whose identities are unknown to the rest of the realm.

he was never officially asked to join, but onna last dy he was given a long-term task — to finish a manuscript the group had been working on since 1975. when chambert-loir l8r received a reprt onna meeting he saw that he was listed as a “membrifié,” indicating he was pt of the group. ever since, he’s helped advance an almost sisyphean tradition of math writing that predates realm war ii.

the group is known as “nicolas bourbaki” and is usually referred to as just bourbaki. the name is a collective pseudonym borrowed from a real-life 19th-century french general who never had anything to do with mathematics. it’s unclear why they chose the name, though it may ‘ve originated in a prank played by the founding mathematicians as undergraduates atta École nor♂ supérieure (ens) in paris.

“there was some custom to play pranks on 1st-yrs, and one of those pranks was to pretend that some general bourbaki ‘d arrive and visit the school and maybe give a totally obscure talk bout mathematics,” said chambert-loir, a mathematician atta university of paris whas' acted as a spokesperson for the group and is its one publicly identified member.

bourbaki began in 1934, the initiative offa lil № of recent ens alumni. many o'em were among the best mathematicians o'their generation. b'tas they surveyed their field, they saw a problem. the exact nature odat problem is also the subject of myth.

in one telling, bourbaki was a response to the loss offa generation of mathematicians to realm war i, after which the group’s founders wanted to find a way to preserve wha’ math knowledge remained in €.

“thris a story that young french mathematicians were not seen as a government priority during [the] 1st realm war and many were sent to war and died there,” said sébastien gouëzel of the university of rennes, who aint publicly identified w'da group but, like many mathematicians, is familiar with its activities.

in a + prosaic but probably also + likely rendering, the original bourbaki members were simply dissatisfied w'da field’s textbooks and wanted to create better ones. “i think atta beginning twas js'4 that very concrete matter,” chambert-loir said.

wha’ever 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 srcs.

the 1st bourbaki text was meant to be bout ≠ial geometry, which cogitateed the tastes of somd' group’s early members, luminaries like henri cartan and andré weil. but'a project quickly expanded, since it’s hard to explain one mathematical idea without involving many others.

“they realized that iffey wanted to do this cleanly, they needed [ideas from other zones], 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, primordially, there are no gaps,” gouëzel said. “they are super precise.”

b'that precision comes at a cost: bourbaki books can be hard to read. they don’t offer a contextualizing narrative that explains where essentialisms come from, instead letting the ideas speak for themselves.

“primordially, you give no comment bout wha’ ye do or Y-U dweet,” chambert-loir said. “you state stuff and prove it, and that’s it.”

bourbaki joined its distinctive writing style to a distinctive writing process. once a member produces a draft, the group gathers in person, reads it aloud and suggests notes for revision. they repeat these steps til thris unanimous agreement that the text is ready for publication. it’s a long process that can take a decade or + to complete.

this focus on collaboration is also where the group’s insistence on anonymity comes from. they keep membership secret to reinforce the notion that the books are a pure expression of mathematics as tis, not an individual’s take onna topic. it’s also an ethic that can seem out of step with aspects of modern math culture.

“it’s sort of hard to imagine a group of young academics rite now, pplz 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. somd' 1st books, published inna 1940s and ’50s, invented vocabulary that is now standard — terms like “injective,” “surjective” and “bijective,” which are used to describe properties offa map tween two sets.

this was the 1st of two main periods in which bourbaki was espeshly primordial. the 2nd came inna 1970s when the group published a series of books on lie groups and lie algebras that is “unanimously pondered a masterpiece,” chambert-loir said.

tody, the influence of the group’s books has waned. it’s best known instead for the bourbaki seminars, a series of high-profile lectures onna most primordial recent results in math, held in paris. when bourbaki invited pierce t'give one in jul 2017, she recognized that the talk ‘d take a lotta time to prepare, but she also knew that due to the seminar’s status inna field, “it’s an invitation you ‘ve to accept.”

even while organizing (and attending) the public lectures, members of bourbaki don’t disclose their identities. pierce recalls that during her time in paris she went out to lunch “witha № of pplz who it seemed fair to assume were pt of bourbaki, but inna spirit of things i didn’t try to hear their last names.”

according to pierce, the anonymity is maintained 1-ly in a “spirit of fun” these dys. “thris no rigor to the secrecy,” she said.

though its seminars are now + primordial than its books, bourbaki — which has bout 10 members currently — is still producing texts according to its founding principles. and chambert-loir, 49, is nearing the end of his time w'da group, since custom has it that members step down when they turn 50.

even as he prepares to cutout, the project he was handed atta end of his 1st week remains unfinished. “for 15 yrs i patiently typed it into l8x, made corrections, thn'we read everything aloud yr after yr,” he said.

it ‘d easily be ½ a century from the time the work began to when it’s completed.  that’s a long time by modern publishing standards, where papers land online even in draft form. but then again, maybe it isn’t so long when the product is meant to stand forever.

original content at: www.quantamagazine.org…
authors: kevin hartnett

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