A mathematician’s autobiography

I recently came across Laurent Schwartz’s autobiography, published in French in 1997, and in English in 2001. The book is hard to read, for various reasons, and has not become well-known; but there is much to be extracted from it.

Schwartz was one of the foremost mathematicians of the middle of the 20th century, a Fields Medallist in 1950. He was also a Trotskyist from when he was shocked by the Moscow Trials, in 1936, at the age of 21, until 1947. He lived through World War 2 in France, doubly at threat because he was both a Jew and a Trotskyist, escaping capture by the Nazis only by a hair’s-breadth on at least two occasions. He was an energetic left activist all his life, often cooperating with Trotskyists in campaigns against France’s war in Algeria, the US war in Vietnam, the USSR’s war in Afghanistan, etc.

“Mathematical discovery is subversive and aways ready to overthrow taboos”, he writes, summing up the connection he sees between the different strands of his extraordinary autobiography.

His own main discovery, the theory of “distributions” (generalised functions), he explains as a matter of finding a coherent mathematical theory to generalise and cover what had previously been slapdash mathematical expedients – which “worked”, but looked as if they shouldn’t – by the physicists Oliver Heaviside and Paul Dirac.

He is critical of mathematicians who disdain that sort of improvisation. At the same time, he was a member of the Bourbaki group of notoriously “pure” and abstract mathematicians. He is critical of Bourbaki’s neglect of applied mathematics and of probability theory, but regards the group as having doing much good.

Schwartz argues that the Bourbaki project would have been impossible except that André Weil, one of its founders, had gone to Germany to study with Emmy Noether and others in the 1920s, when most French mathematicians were trying, for chauvinist reasons, to ban Germans from international mathematical conferences.

The Bourbaki group produced 19 books, over many years, as a systematic rewriting of large areas of mathematics in the way that Noether and her colleagues had rewritten algebra.

It was an extraordinary procedure, maybe the only example in history of important books being produced in a more-or-less planned way by a committee. Each area of mathematics was successively named as the subject for a book. (There were many arguments about the order).

One member of the group would then write a “zero-th” draft of a book. The draft would be “completely demolished” in the group’s stormy, rowdy monthly meetings. The main organiser of the group once it got going, Jean Dieudonné, whom Schwartz describes as doing mathematics full-tilt 18 hours a day, every day, would threaten to walk out, or actually walk out, at almost every meeting.

Then another member would write another draft. Then another, another… until “around the seventh or eighth version”, the group finally conceded that a draft was ready to publish under the authorship of the fictitious “Nicolas Bourbaki”. The result was not a textbook, nor a report of research – members of the group wrote their own textbooks, and research reports, separately – but an attempted model of how the particular area of mathematics could be systematised and generalised.

The project never achieved its stated goal. Pure mathematics was expanding much faster than the group’s attempts to systematise it, and the group never tried to integrate applied mathematics. But Schwartz is surely right to say that Bourbaki changed the whole style of mathematics.

Schwartz describes himself as having an “enormous” memory, but an extremely poor visualisation of space. He is, he says, chronically unable to remember routes and directions, and equally: “I visualised almost nothing when studying geometry”.

He makes no comment about a possible connection between this unusual mindset and the opposition of the whole Bourbaki group to the use of diagrams in mathematics (as obscuring general concepts with too-specific illustrations), an opposition which has arguably had a negative effect on mathematical development. I wonder.