A Scene Change
When I turned 16, I received a pamphlet in the mail from Simon’s Rock College with this tagline: “What if you’re ready for college now?” I wasn’t sure about college, but I was tired of taking what seemed like the same math over and over: Algebra I, Algebra II, Algebra III, etc. It felt like running in place, and I wanted a change of scene. I visited Simon’s Rock and asked students there about the math classes. Were they all the same? “No, but…” But what? “But the books are all by Serge Lang—a bit of a Bourbakist,” I was told. Even basic mathematics? Calculus? Linear algebra? I was referred to Lang’s Basic Mathematics, First Course in Calculus, and Linear Algebra. I had a look: it turned out I was ready for college.
Not content to be a high school dropout, that fall I tried courses in everything I couldn’t take in high school, starting with calculus. Lang’s Ph.D. student Allen Altman was my teacher. He told me Lang was famous for saying that a problem was trivial, especially when the folks around him found it far from obvious. In a book he’d write “trivial” and wash his hands of it. Was he a good teacher? The keys to good teaching include figuring out how to make the ideas shine through and keeping your audience in mind. Altman suggested that Lang sometimes kept you in mind by yelling at you and made ideas shine through by throwing chalk. I was glad not to be assigned audiobooks.
The Most Prolific
In the spring I took differential equations, much simplified by complex numbers and Euler’s famous formula . “Who was Euler?” I wanted to know, and found that he was “the most prolific mathematician of all time.” To this day, it is said that people wandering around the oldest libraries of Europe continue to find Eulerian propositions slipped in between the pages of books which haven’t seen the light of day in centuries. I couldn’t help but think to myself, “Maybe, but I doubt he could be as prolific as Lang!”
When the final tally is in – and it nearly is – it would appear that Euler and Lang are roughly tied for the prize of “greatest mathematical output for any individual.” Euler continued to produce mathematical theorems even after he’d gone blind in one eye – King Frederick the Great called him a “cyclops” – thanks to his scribe. It’s said that the scribe’s first assignment was to accept an algebra book by dictation and do all of its exercises so that he might be able to arrange the symbols properly in Euler’s further works. On the other hand, there are all kinds of stories about Lang’s ability to type with phenomenal speed. I’ve heard that during the Korean War he received secret keyboarding training from the military that allowed him to type faster than any man then alive (but not, in fact, any woman, and I hope you take such tall tales with a grain of salt).
A Master’s Company
Above I said “any individual” because there is one prolific mathematician, Bourbaki, whose work may actually be the collective labor of a secret society. An article written by several mathematicians explains that Bourbaki is “their Master,” that they are his “disciples,” and that they “chose to publish their Treatise under the name of their Master.” The American mathematician, Boas, in an article on Bourbaki appearing in the Encyclopedia Britannica, claimed “Bourbaki” was just the pseudonym used by a covert ring of young mathematicians and upstarts—a group of agitators claiming legitimacy under the aegis of a nonexistent Master. Why would they want to do that?
In the 1930s, the disparate branches of mathematics were highly disorganized. Some feared that the subject would split into sharply distinct subdisciplines just as natural philosophy had split into chemistry, geology, physics, and biology. In Germany, however, young Bartel van der Waerden’s Moderne Algebra pulled number theory and algebra together by recasting them as the study of whole structures, like number systems, rather than individual equations. Roughly speaking, the Bourbakists’ program was based on the hope that a similar but more ambitious reorganization would unify the entire field of mathematics. Yet who would listen to what these young mathematicians and upstarts had to say? The senior French mathematicians would not endorse this revolutionary project, so the group apparently invented a yet more senior mathematician, a Master from Poldavia related to a deceased French general. As Poldavia is impossible to locate on a map, this made fact-checking difficult.
Both Legend and Joke
Whether or not the Bourbakists created their Master, they have been most vigorous in defending poor Bourbaki against accusations of nonexistence. Annoyed with Boas for his Britannica article, Bourbakists tried to get even by circulating the rumor that Boas didn’t exist, that B.O.A.S. was the acronym of the editorial board of Mathematical Reviews, of which Boas really was a member. Moreover, some Bourbakis from Greece have stepped forward to claim him as a relative – if you have relatives, don’t you exist? I’ve heard that on his deathbed the Frenchman Jean Dieudonné admitted that he’d been the one who’d shepherded Bourbaki’s greatest work, the monumental Elements of Mathematics, to the presses. So influential has this work been that almost all modern mathematical notation and terminology can be found in or traced back to it. Dieudonné insisted to the end, however, that Bourbaki had shared his “notes” with him, and that like Euler’s scribe, what he’d been taught was the bare minimum to avoid wrecking that which is truly sublime.
At the Joint Mathematics Meetings in Boston of January 2012, I asked some of my French colleagues whether it was true that there was a street named after Bourbaki in Aquitaine, even though he presumably had no presence there or anywhere. I learned that this was indeed true, but one hedged as to whether Bourbaki really had “no presence.” He said that Bourbaki “had an office” in his own institution, though he was frequently absent and there was some concern as to his health. I shouldn’t reveal this individual’s identity, in case he was a Bourbakist himself (the disciples are sworn to keep their identities secret until middle age). Lang, an American, was briefly involved as a Bourbakist, which is surprising since all of the others were French. This fact appears not to be widely known among the mathematical public, however – or, conceivably, is the origin of the following well-known joke:
Q: Why did Bourbaki stop publishing?
A: He found out that Serge Lang was one person!
For more stories (with pictures) about Bourbaki, check out Bourbaki: A Secret Society of Mathematicians by Maurice Mashaal.