Dinner conversation

The main argument of yesterday's conversation seems to me worth of a separate post, if nothing else because I think it was some remark of mine on this topic that induced the two senior colleagues in question to invite me to join them for dinner.

Mathematics is in one significant respect very different from most other scientific subjects: namely, most of the research done in the past decades is still correct and applicable. However, mathematical fashion changes, and the language of mathematics moves on: so, in time a natural selection occurs, and results that are perceived as important get incorporated in standard reference books, and the rest gets forgotten and sleeps in many a library shelf. In particular, history tends to be obliterated, and young people know definitions and theorems without knowing who first introduced or proved them.

However, in the period 1960-1970 my field had an incredible change of pace; tons of new ideas where introduced at a speed that most researchers of the time, not to mention a publishing system based on typewriters, had big problems keeping pace with. I am not a historian of science, but my impression at 40 years of distance was that the key point in this gargantuan enterprise was one person: Alexander Grothendieck. His ideas have changed the face of algebraic geometry as we know it, and lie at the base of many current famous results, most notably Fermat's last theorem.

My comment was that a large part of the work done in those years seems to have been forgotten and not passed on to the younger generations as it should have. I started thinking about this when Faltings, at the beginning of a course on algebraic stacks, explained that stacks were as important as schemes, but the generation that had had to digest schemes had refused to deal with stacks as well. In the same weeks, I was working my way through as much as I could understand (i.e., not much) of Grothendieck's mathematical testament, Récoltes et Semailles. There he expresses precisely the same viewpoint, namely that a lot of his work has been factually hidden from the eyes of the average working mathematician.

Whether this is due to someone's agenda (Grothendieck makes quite a few high-level names, which has made Récoltes et Semailles unpublishable) or it just happened, it is definitely time to go back, and retrace our steps. An impressive result in this direction is the re-edition of SGA1 and SGA 2 by the Société Mathématique de France; it is a LaTeX typeset and corrected version of some fundamental, typewritten notes from the early sixties, the kind of thing that an old enough library would have, but was difficult to get for private use. Now it is affordable to everyone (especially members of the SMF).

What I think is very significant is the way the typesetting was done: namely, a number of mathematicians from all over the world volunteered hours and hours of work to input, and correct the .tex file. It is, literally speaking, a labour of love.

My point yesterday was that we need more of this, both in the sense of re-editing (and making available in electronic form) old books, and also in the sense of translating the old books in modern mathematical language, so that everybody can read them. We should use the possibilities of the new media to produce a document which is readable at different levels; a short one for the experts, but one where one can just click on an unfamiliar name to read its definition, and on a step of the proof to find a more detailed version, including the details known to the experts.

And most of this has to be labor of love; but if we want it to happen, we have to start giving at least some credit to people who do the work. I am thinking of all the mathematicians who already are tenured, and who therefore are not any more under desperate pressure to publish. It shouldn't look like a hole in the cv if in one given year you have no paper but a beautiful edition/exposition of old, inaccessible stuff.

Let me finish with a beautiful sentence I heard at lunch not very many days ago by a colleague who is a bit older than me: "I could of course keep writing paper after paper. But there is so much beautiful mathematics out there that I don't know yet, and I want to read at least some of it before I die". Me too.