Amir Aczel’s book on the Bourbaki group of mathematicians (*The Artist and The Mathematician*) is well-written and readable, but, I think, almost completely uninformative and even misleading about Bourbaki’s maths.

“Nicolas Bourbaki” was a collective pseudonym for a group of French mathematicians active mainly between 1934 and the 1960s. (The group still exists, but has published little in recent decades, and it has had some non-French members).

Aczel’s book tells us mostly about the lives of two of the members of Bourbaki, André Weil (the founder of the group) and Alexander Grothendieck (the group member most famous for his individual research).

They’re colourful enough. Weil, brother of the philosopher and sometime revolutionary socialist Simone Weil, nearly got shot as a Russian spy in Finland in 1939. He was saved only by the Finnish chief of police happening to mention to Rolf Nevanlinna, then a colonel in the Finnish army, that Weil was to be shot the next day, and Nevanlinna persuading the police chief that Weil could instead be deported.

Though Aczel doesn’t tell us, Nevanlinna was himself a well-known mathematician (one who, unusually in the 20th century, made his best discoveries while still working as a school teacher). He was also a right-winger, former member of the White Guard in the Finnish Civil War, and Nazi sympathiser, but his respect for Weil as a mathematician overrode those views.

Weil, Jewish like Grothendieck and other Bourbaki members like Laurent Schwartz (a one-time Trotskyist and lifelong anti-Stalinist socialist), had a difficult time in World War Two France, but eventually escaped to the USA.

Grothendieck was born to anarchist parents, was a lifelong leftist himself, had to survive much of World War Two as a young teenager separated from both his parents, was largely self-taught in maths until he got to study with Schwartz and Jean Dieudonné (another Bourbaki member) in Nancy at the age of 21, revolutionised algebraic geometry in the 1950s and 60s, and then mostly stopped publishing mathematics after 1970, becoming a complete recluse from 1991 to his death in 2014.

In other academic fields, having your name on a publication is a priority. Not so much in mathematics. All the members of Bourbaki also published work under their own names, but they refused to credit their “Bourbaki” books to anyone other than the fictional “Nicolas Bourbaki” of the fictional “University of Nancago”.

A more recent collective author in mathematics is D H J Polymath, used to report results of the collective “Polymath projects” initiated by Tim Gowers.

Polymath publishes research. Bourbaki didn’t really. Grothendieck’s work in algebraic geometry, for example, was published by him, not Bourbaki.

The Bourbaki group started in the mid-1930s as a collective project to write a new textbook for French universities on analysis (which means, roughly, calculus done properly and extended). They wanted to replace the textbook then most used because it was based on lecture notes from the 1890s, which, as we’ll see, made it considerably out of kilter by the mid-1930s.

Soon the project turned into something different from a textbook. As Claude Chevalley, one of the Bourbaki mathematicians (and, again, politically an anarchist), has said in an interview, the nine books published between 1939 and 1967 (or, at least, published in part: they were published in sections or “chapters”, sometimes decades apart) were “useless for teaching”.

They were, in the end, more like “fair copies” of existing theory – writings-out made as rigorous and orderly and concise as possible. And as unified as possible: the overall title for the book series was “Elements of mathematic”, the word “mathematic”, singular, being chosen over “mathematics”, plural.

That was quite something. The books became an important point of reference. I can remember thumbing through the Bourbaki volumes with some awe in the library of the Cambridge Department of Pure Mathematics and Mathematical Statistics in the late 1960s. (I never systematically studied them).

It was clear from early on that the series would never achieve its aim of unifying “mathematic” into a single, systematic body of exposition. There was never any thought of covering applied mathematics or statistics, or indeed such branches of pure mathematics as probability theory and number theory. New branches of mathematics were developed, or old branches revolutionised, quicker than Bourbaki could possibly systematise. (There was never a Bourbaki book on algebraic geometry, for example).

Oddly, the area of more recent mathematics which shifts attention most towards structures and relations rather than particular mathematical objects – category theory, developed in the USA in the 1940s by Saunders Mac Lane – was not covered at all by Bourbaki, though Grothendieck used it a lot in his individual work in algebraic geometry.

Aczel argues that the Bourbaki books nevertheless introduced the ideas of rigorous reasoning and of “structures” into mathematics, and from the late 1940s it drove the “structures” idea to become also influential in other fields like linguistics and anthropology.

Similar arguments can be found in other accounts of Bourbaki. I think they are seriously misleading.

Rigorous reasoning in mathematics goes back to Euclid. Yes, some hidden assumptions in Euclid’s reasoning have been found since then. That means the concept of rigour has been refined and sharpened, not that it has been developed for the first time.

In the 17th and 18th centuries, especially, as calculus developed, so also developed a big range of mathematical arguments which broke new ground but which the mathematicians of the time could not yet make rigorous.

That making-rigorous was not, however, done by Bourbaki. It was done primarily by Weierstrass, Dedekind, and others in the late 19th century.

Their efforts spurred Hilbert (in 1899) to examine the foundations of geometry (now widened to include non-Euclidean geometries) and others, like Peano, Frege, and Russell, to investigate rooting the theory of numbers in the theory of sets. After all, what is a number? By the late 19th century that question had been probed and prodded enough to make it problematic.

It turned out that the theory of sets, which apparently assumes very little, has pitfalls; and, in any case, bizarrely but provably, no system of strict axiomatic reasoning strong enough to include the basic theory of counting numbers can generate proofs of every true statement in it.

But there are workarounds and, as we’ll see, they were used in *algebra*. In 1934, when Bourbaki started, no-one had reworked the great body of theory in *analysis* which had been developed in fits and starts over the previous couple of centuries into a coherent system on the basis of those workarounds.

Mathematics had started to turn to thinking about more general structures, as well as about notionally stand-alone particular “objects” – triangles, circles, functions, series, etc. – from the early 19th century at latest.

Over the centuries mathematicians had calculated formulas for solving quadratic, cubic, and quartic equations. Evariste Galois showed that there is *no* formula in square, cube, 4th, 5th… roots for equations with powers as high as x^{5} or higher, not by calculation but by investigating the structural properties of the *systems* of roots of those equations.

Mathematicians had used complex numbers, a bit shamefacedly, as ad hoc gambits, for example in solving cubic equations (even when all the roots finally calculated are real). Carl Gauss started the study of complex numbers as a whole new *number system*.

The Bourbaki mathematicians didn’t deny that history. In an introduction to one of their books, they wrote: “One could be tempted to say that the modern notion of ‘structure’ had been gained in substance by 1900; in fact, it would take about another 30 years of development for it to appear in full light”.

At the end of those 30 years Bartel van der Waerden published a book, *Modern Algebra*, based on the work of the German mathematicians Emmy Noether and Emil Artin, which, as another mathematician would comment, “forever changed for the mathematical world the meaning of the word algebra from the theory of equations to the theory of algebraic structures” – that is, structured sets: groups, rings, ideals, fields, vector spaces, etc. So van der Waerden’s book starts with chapter 1 on sets, chapter 2 on groups, chapter 3 on rings, and so on.

Bourbaki tried to do the same transformation on analysis and closely-related areas of mathematics. Thus, the textbook on *Foundations of Modern Analysis* by Jean Dieudonné (a leading member of the Bourbaki group), which I used when studying at university, has chapters on sets, metric spaces, normed spaces, Hilbert spaces, and spaces of continuous functions, before it even starts discussing differentiation.

Aczel misses two dimensions. French mathematicians were in a plight in the 1930s not only because their numbers had been thinned by World War One, and not only because of the inherent difficulty of integrating and systematising the pell-mell development of new theories, but because of chauvinism.

Germany was the centre of world mathematics from the early 19th century through to Hitler’s seizure of power in 1933. But for a period after World War One, the French government, and some French mathematicians, had a hostile attitude to German mathematics.

Weil and Chevalley were exceptional because they went to study at German universities (Weil at Göttingen, where Noether was working, in 1927, and Chevalley at Hamburg, where Artin was working, in 1931-2: Chevalley earned a mention in later editions of van der Waerden’s *Algebra*, p.132 of the 3rd English edition, for a result of his published in Hamburg in 1935).

In some part, Bourbaki was about the introduction of “German” methods into “French” mathematics – at a time when mathematics in Germany itself was being destroyed by Hitler. Many outstanding German mathematicians were Jewish or democratic-minded, and fled from Germany, usually to the USA, after 1933. Noether and Artin did, for example.

Bourbaki introduced a “lurch” into French mathematics, or at least into part of it, towards discussing mathematical topics at the most abstract and generalised level possible. That has its place, but also its downsides.

Miles Reid, in his textbook on algebraic geometry, explains some of the downsides: “A whole generation of students (mainly French) got themselves brainwashed into the foolish belief that a problem that can’t be dressed up in high-powered abstract formalism is unworthy of study, and were thus excluded from the mathematician’s natural development of starting with a small problem he or she can handle and exploring outwards from there”.

More generally, Ron Graham, Donald Knuth, and Oren Patashnik were impelled to write a textbook entitled *Concrete Mathematics* in counterpoint to the always-more-abstract approach spearheaded by Bourbaki. They argued:

“Abstract mathematics is a wonderful subject, and there’s nothing wrong with it. It’s beautiful, general, and useful. But… the goal of generalisation had become so fashionable that a generation of mathematics had become unable to relish beauty in the particular, to enjoy the challenge of solving quantitative problems, or to appreciate the value of technique”.

Aczel’s argument that Bourbaki’s emphasis on structures shaped other fields of study rests heavily on the fact that André Weil wrote a mathematical appendix in the anthropologist Claude Lévi-Strauss’s 1949 book (once celebrated, but now out of fashion), *The Elementary Structures of Kinship*; and Lévi-Strauss called his approach “structuralist”.

But the appendix is quite elementary group theory. Lévi-Strauss asked Weil to do it because he, Lévi-Strauss, was a professor, and could ask another professor to do such a thing; but in fact a diligent first-year university maths student could have written it.

I doubt that “structuralist” approaches in mathematics, in anthropology, and in linguistics had much in common beyond vague analogy, let alone that they were all driven by the Bourbaki orientation to “structures”.

Below: the Bourbaki group talking about mathematics over lunch, 1951

Below: the contents of the first few Bourbaki books