Book Description

The Book As we have seen, Grothendieck is the author of a considerable body of mathematical work. But he is also the author of significant literary works. Among them is R&S, which was published by Gallimard in January 2022 after having been widely distributed on the internet since Grothendieck first wrote the text in 1986. Amounting to more than 1,900 pages, the book deals with many subjects: the author’s journey as a mathematician, his passions, his illusions and disillusions, the process of creation, and a thousand other topics. It also includes long passages on Yin and Yang, feminine and masculine ways of doing mathematics, the mother, the father and child, dreams, and so on. A large part of the text is devoted to a revelation he is said to have experienced in 1976 and a long period of meditation that followed. It is a kind of self-analysis tinged, it has to be said, with a certain degree of paranoia. A recurring theme is the sense of betrayal he felt toward his former students, which is manifested in his work being ignored and forgotten. The words “funeral,” “deceased,” “hearse,” “massacre,” and “gravedigger,” and so on, quickly become omnipresent after their appearance in the table of contents. More generally, the book denounces a loss of ethics among the entire mathematical community. Grothendieck explains to the reader that mathematics “was better before”—that is, prior to 1960—as if the older generation was irreproachable! In fact, on the contrary, it can be said that mathematicians have become much more honest since the 1990s. The source of this miracle has a name: arXiv. It is now becoming ever more difficult to appropriate the ideas of others, although, of course, it is still possible to some degree. The institution of mathematics itself has also been greatly improved, or at least has been greatly transformed. The system of mandarins that dominated French mathematics until the 1970s, from which Grothendieck did not experience any difficulties and about whom he does not say a word, has practically disappeared. Grothendieck, who is very self-critical throughout the text, sometimes ponders whether he might have been arrogant or even contemptuous of those around him during his heyday in the 1960s and 1970s. Despite these concerns, it is clear that he cares little about ingratiating himself with his readers. Instead, he offers a book of more than 1,900 pages, while in response to a question about the IHES library in its early days, he remarks: “We don’t read books, we write them!”18 R&S contains many contradictions that are only partly corrected by a series of Notes—some of which, despite being of particular importance, are not included in this new edition. Addressing these contradictions properly would undoubtedly have required the text to be completely rewritten. Grothendieck is not paralyzed by any sense of false modesty: The thing that struck me is that I do not remember having known, even from the allusions of friends or colleagues who are better versed in history, of a mathematician apart from myself who contributed such a multiplicity of innovative ideas, not more or less disjointed from one another, but as part of a vast unifying vision (as was the case for Newton and for Einstein in physics and cosmology, and for Darwin and for Pasteur in biology).19 Elsewhere, he writes: “It would seem that, as a servant of a vast unifying vision born in me, I am ‘one of a kind’ in the history of mathematics from its origin to the present day.”20 Although the writing style is not lacking in inspiration, it is nonetheless uneven and sometimes—deliberately—familiar. Grothendieck is not le duc de Saint-Simon. The following analysis will focus only on the content concerning mathematics and the world of mathematicians. In the text, Grothendieck complains at length that his ideas have been plundered by his former students without reference to their master or that they have simply been erased and forgotten. These assertions are not always supported by solid arguments or precise references. But, above all, it is the nature of discoveries to be trivialized and their author forgotten, and all the more so when the underlying ideas are often, in hindsight, obvious. Grothendieck’s reproaches are addressed to all his pupils, and particularly to Deligne—whose name is almost always preceded by the words “my friend,” insinuating “my former friend”—and to Verdier. It is quite possible to imagine that Deligne was only lightly involved with Grothendieck’s authorship of the motives or that the “Verdier duality” already mentioned could just as well be called the “Grothendieck duality.” But, otherwise, everyone knows that it was Grothendieck who invented schemes, motives, Grothendieck topologies, topoi, and, above all, that he imposed the functorial point of view via the six operations and the derived categories. Everyone knows that it is thanks to the machinery devised by Grothendieck that Deligne was able to prove André Weil’s last conjecture. In support of his claims about the total loss of ethics in the mathematical community from the 1970s onwards, Grothendieck’s entire argument is based on the unique testimony of one and only one mathematician who came to see him several times at his home in the countryside. It is common practice in ethnology to rely on an informant from the group being studied and who speaks the language. The problem is that the informant may not always be all that reliable and can, in fact, say anything. Here it is an even worse situation, since the informant declares himself to be the first person affected by the story he is going to tell, namely the Riemann–Hilbert (R–H) correspondence.