The mathematician who doesn’t exist

One of the world’s most important mathematicians has been working for nearly a century, producing dozens of books totaling thousands of pages that have served as a benchmark for the entire field. His name is Nicolas Bourbaki and he does not exist.

Bourbaki is the pseudonym of a secret society of mathematicians. Created in France in 1934, the group began with a simple goal: to update mathematics textbooks and make them more relevant to a contemporary audience. Instead, he created a whole new way of writing math that will make waves for decades to come.

Initially, the group thought their work would be about a thousand pages long and take six months. In 1935, Bourbaki had decided to write a series of six books, each building on the previous one to “provide a solid basis for the whole of modern mathematics”, as he would later indicate in an explanatory introduction. The group thought it would be over 3,000 pages and would be completed within a year. They got the first item mostly correct, and the second item very wrong.

Despite the intention of reading the books (which ultimately consisted of several physical volumes) in order, Bourbaki’s first published text, in 1939, was the final chapter of what would become the first book, Set theory. From there, the group evolved, publishing various chapters of other books over the years and only returning to Set theory in 1954, finally completing it in 1970. The entire work was eventually labeled Math elementswith the unusual singular intended to emphasize the work of mathematicians as a coherent whole. The six books were not finalized until the 1980s, with a final total of nearly 4,000 pages – although by then Bourbaki continued to publish new books as the scope of the initial project expanded further.

This anarchic pace of publication is due to Bourbaki’s unique way of working. The initial group consisted of half a dozen young mathematics professors, including André Weil, who would have considerable influence on number theory and algebraic geometry. Most were former students of the École Normale Supérieure in Paris, France, and it was a prank from their university years involving an incomprehensible “Bourbaki theorem” that inspired the group’s name.

This prankster attitude was the key to group cohesion. The meetings were chaotic and alcohol-fueled, often devolving into arguments and lewd jokes. One member would produce a suggested text and read it, line by line, so the rest of the group could critique and challenge it. Another member would then produce a revised text, and the process would continue until there was unanimous agreement. It’s no wonder it took so long, with the average chapter taking 10 years to produce. Bourbaki members were asked to retire when they reached the age of 50, and others were recruited to replace them. So it was a multi-generational mathematical effort.

An eternal problem

But what was Bourbaki really doing? Unlike the way it was produced, Bourbaki’s work was sober and rigorous to an excess. Set theory aimed to build a foundation that could tackle a perennial problem at the heart of mathematics, namely that the mathematical objects and ideas that concern mathematicians are independent of human language or symbols.

To understand why, think of the word “addition” or the “+” symbol. These have a completely arbitrary relationship to the actual underlying mathematical concept – we could use any string of symbols to denote addition, as long as we agree on what it means. On the other hand, addition has a strict and intrinsic relationship with subtraction, because one reverses the other and this is true whatever the name we give them.

In practice, labeling mathematical concepts is not a problem because mathematicians have conventions for a standard correspondence between concepts and words or symbols, but in principle there is the possibility of contradiction or disagreement.

Bourbaki was not the first to attempt this type of formalization (I recently wrote about some early efforts here) but he was perhaps the most pedantic. For example, the number 1 is carefully defined in a footnote on page 158 of Set theory. Bourbaki writes that “the symbol ‘1’ should of course not be confused with the word ‘one’ in ordinary language”, but rather should be considered equal to the following definition:

τZ ((∃u)(∃U)(u = (U, {∅}, Z) and U ⊂ {∅} × Z and (∀x)((x ∈ {∅}) ⇒ (∃y)((x, y) ∈ U)) and (∀x)(∀y)(∀y’)(((x, y) ∈ U and (x, y’) ∈ U) ⇒ (y = y’)) and (∀y)((y ∈ Z) ⇒ (∃x)((x, y) ∈ U))))

Don’t panic. I can’t attempt a full analysis of this here, although a very high-level explanation is that ∅ is a set (a mathematical term for a collection of objects) and that set contains zero objects, making it “the empty set”. From there, 1 is defined as {∅}, the set containing an object, that object being the empty set. You can read more about this in a previous column.

What’s incredible though is that this jumble of symbols actually hides a much broader formal definition, with each squiggle carefully and excruciatingly defined based on the previous text of the book, using only the symbols τ, ∨, ¬, ☐, =, ⊂ and ∈. It is worth saying that Bourbaki never writes them down in full – the footnote estimates that this would require tens of thousands of symbols for this definition. This turns out to be a significant underestimate, with later mathematicians calculating that writing the full expression for the number 1 would require over 4.5 billion symbols, or perhaps 2,409,875,496,393,137,472.149,767,527,877,436,912,979,508,338,752 092,897 symbols, depending on how strict you want to be.

Clearly, moving away from such intense formalization is necessary if mathematicians are to actually get work done, and Bourbaki admits as much – while still insisting that using shortcuts like “1” or “one” is a “misuse of language.” By establishing the rules, Bourbaki gave mathematicians permission to break them.

The problem with New Math

So what did all this actually achieve? On the one hand, this allowed Bourbaki to unify mathematics as a singular entity. If, in theory, terms and concepts from two different branches of mathematics can be described using the same basic symbols, this provides a rigorous basis for moving from one to the other. In practice, no one does this, but it puts mathematics on firmer philosophical ground. And decades later, Bourbaki’s approach is proving surprisingly influential, as mathematicians explore the use of computer-aided formalization to verify proofs produced by artificial intelligence. Bourbaki also introduced many concepts and symbols (∅ for the empty set, for example) which remain used by mathematicians today. More broadly, the Bourbakian writing style continues to influence modern mathematics textbooks.

Bourbaki was not without his detractors, however. Like the publication of Math elements Following this, some mathematicians rebelled against the group’s insistence on pedantic rigor. More bizarrely, Bourbaki inspired a disastrous attempt to rethink the way mathematics is taught in schools. First appearing in the late 1950s in France and then spreading to the United States and other countries, the “new mathematics,” as it was called, sought to abandon traditional teaching tools like multiplication tables and instead adopt a rigorous mathematical approach based on set theory and based on the teachings of Bourbaki. The goal was to understand the general idea of ​​multiplication, for example, rather than memorizing specific facts like 3 × 4 = 12.

New Math was widely considered a disaster. Parents did not understand what their children were being taught, and neither did teachers in many cases. A successful book, Why can’t Johnny addserved as a stinging rebuke, and by the end of the 1970s New Math had been largely abandoned. The 1970s were also bad for Bourbaki on another front, as the band was forced to wage a legal battle over copyright and royalties with its publisher.

Nevertheless, Bourbaki remains active today, publishing two new book chapters this year, although, as is tradition, the authors behind them remain secret. In a way, the secrecy allows the mathematicians to treat Bourbaki like a slightly embarrassing uncle: everyone is happy that he is there, doing work that no one else wants to do, but at the same time the mathematicians are relieved not to have to invite him to dinner.