El País, La Nación, Frankfurter Allgemeine, The New York Times. Still in bed, I briefly checked those newspapers on my laptop, as I do every morning: a needful reminder that dreams are nothing but, that the world is ugly and the people are sad. This past November 14 the world seemed no different from any other day; then I turned to Le Monde, and everything was suddenly changed: « Alexandre Grothendieck, le plus grand mathématicien du XXe siècle, est mort » http://www.lemonde.fr/disparitions/article/2014/11/14/le-mathematicien-alexandre-grothendieck-est-mort_4523482_3382.html
I first heard that Alexander Grothendieck was the greatest mathematician alive from the lips of one of his teachers, Jean Dieudonné. This was in Bariloche, in 1961, on the shore of Lake Nahuel-Huapí in the Patagonian Andes, where I spent half a year teaching math to the physics students at what is now called the Instituto Balseiro. The previous year I had graduated from the University of Buenos Aires as a licenciado en matemáticas, roughly equivalent to a master’s degree; also the previous year, the region had felt the marginal effects of the Valdivia earthquake, the strongest ever registered anywhere, and all boats in the lake had been sunk and all the piers destroyed. Nonetheless, in September or October 1961, in the early spring anyway, M. and Mme. Dieudonné, together with their daughter, who was about my age, ravishingly elegant and pretty, descended from the boat which had sailed from Puerto Montt, in Chile, and, after crossing several lakes, had arrived in Bariloche, in Argentina. I was deputized by the Institute to meet them at the renovated pier, on account of my being a mathematician and of being perhaps the only one around who spoke French.
The Dieudonnés were on vacation. They came to Bariloche because their daughter wished to ski on those famed slopes, which kept her busy, up and down, down and up, until they left. Mme. Dieudonné spent her time otherwise, in tourism or shopping. I had Professor Dieudonné all to myself, and I couldn’t be happier, for he was very kind and willing to talk, and I was eager to learn more about my admired Bourbaki, whose books were then in the apogee of their influence and prestige, having played a significant role in the development of French Structuralism. I organized a talk for the students of the Physics Institute, in which Jean Dieudonné explained to them about the Bourbaki group, and I translated between his French and the audience’s Spanish.
Nicolas Bourbaki is the collective name of a group of young French mathematicians who in the mid 1930s decided to write an open-ended encyclopedia: starting from the axioms, it would gradually embrace most of math, under the title « Éléments des mathématiques » and the subtitle « Les Structures fondamentales de l’analyse ». By the time of my story, six sub-subtitles had been published. Bourbaki’s style was austere and his language aspired to complete unambiguousness, so that for instance after defining the algebraic structure called “group,” which consists of a set G together with an operation x satisfying certain requirements, the author(s) kindly allows us to refer to the group just by the letter G—by an abuse of language. « Par abus de langage » : la petite phrase still resonates in my mind, with its godlike assurance that one could always tell what’s use and what abuse, and that it faithfully voiced the fateful knowledge of right and wrong. Dieudonné was among the handful of founders of the Bourbaki group and, more so than the others, was credited with its logical and linguistic style.
The identity of the members of Bourbaki was kept secret, and membership ceased at age fifty; since Dieudonné was fifty-five he could speak freely about himself and his coevals, though not about the newer generations, which included the thirty-three-year-old Alexander Grothendieck. But Grothendieck, we were told, had resigned his membership, having more important business than working for Bourbaki: he was creating new and fundamental math at such prodigious speed that an amanuensis was needed, and he, Dieudonné, had taken upon himself the full-time job. It is not once every century that a mature, distinguished scientist relinquishes his own career to become the secretary of a former student of his, and it was heart-warming and uplifting to know that Dieudonné’s notion of what was mathematically important proved stronger than any scruples of self regard. “Grothendieck,” he said, “is the greatest mathematician alive.”
When we were alone, Dieudonné asked me what was I teaching the physics students, and I replied that functional analysis, together with some integral equations, the Fredholm alternative, and so on. I had been using Leçons d’analyse fonctionnelle by F. Riesz and B. Sz.-Nagy, a book that was about ten years old and in a style wholly different from Bourbaki, clear and elegant, no doubt, but with no comparable concern for generality.
“When it comes to integral equations,” said Dieudonné in accents that did not invite reply, “the fundamental theorem is that if a topological vector space E on a complete, non-discrete and locally compact valuated field K, is locally compact, then E is of finite dimension. Are you teaching them that?”
I wasn’t, and offered in bland justification that physics students would not climb with me gladly to such peaks of abstraction. Later, alone in my room, I reflected and saw that Dieudonné was right, that viewing those equations whose unknown is a function in the light of general theorems such as the one invoked by him, made everything more clear and easier to place in the vast scheme of mathematical objects. It made everything more beautiful. Which didn’t necessarily mean (I also reflected) that it was pedagogically sound. Indeed, the so-called high-school new math, in which Euclidean geometry had been replaced by set theory, was championed by M. Dieudonné and was, in my opinion, an educational disaster.
I met Grothendieck for the first time early in 1970, in Pisa, the town of Galileo, where he had been invited by Prof. Aldo Andreotti to give a talk. We were five or six in the audience, and I don’t think any of us understood half the whole —certainly I didn’t. Yet this much was clear: the method of studying algebraic geometry problems by looking at the algebraic equations over different fields, including finite ones —while I always limited myself to the real and complex numbers— yields beautiful and unsuspected results. This was the opposite of generalizing just for the sake of generality.
In August 1970 Isabel, our children, and I returned to the U.S., then in the depths of the Vietnam moral catastrophe. As we were driving back to our apartment in Madison, Wisconsin, we heard on the radio that the Army Math Research Center (AMRC) had been blown up by a truck bomb. A graduate student had been killed in the adjacent physics building. The University of Wisconsin campus was in upheaval, one protest following another. There were demands for a Black Studies Department, for the demise of the AMRC and for an end to ROTC recruiting on campus, for North Vietnam (Ho, Ho, Ho Chi Min!), even for the liberation of children from parental tyranny. It may have been that heady and effervescent atmosphere together with the outrage at the U.S. cutting-edge-tech-barbarism in Indochina, or perhaps it was the seeds planted in my mind by a first contact with classic Mediterranean art and culture, or, again, the sudden change from Italian ancient cynicism to American murderous innocence and earnestness—whatever the causes, I became a radical. I became a supporter of Science for the People and, based on my being born and raised in Latin America, a member of the campus Third World Collective: together we read Marx, Engels, Lenin, and Mao.
The AMRC was planning a big meeting, a Symposium on Non-Linear Functional Analysis, in April 1971; in February I wrote a letter, signed also by four of my colleagues in the Math department, to those who were invited, asking them not to attend an event sponsored (directly or indirectly) by the US Army. I sent a copy of the letter to Grothendieck, who replied at some length, expressing an interest in my activities and a willingness to talk publicly about his at my invitation. At that time, having resigned from the IHES (Institut de Hautes Études Scientifiques), where he had, with Dieudonné, done his best mathematical work over the previous decade, Grothendieck devoted much of his enormous energy to the organization and the journal Survivre or Survival (later changed to Survivre et vivre), founded by him, the algebraist Claude Chevalley, and several others who felt that survival had become the most urgent task, superseding that of doing math or indeed any other. After climbing the gradus ad parnassum, the ascending levels of mathematical generality, apparently Grothendieck had discovered that there is the great ocean of mortal, unplatonic Being out there, and that it demands a lower stance, at sea level.
Among the letters I received from Grothendieck there is one I will transcribe in its entirety because it shows quite clearly the sort of temperament he brought to the task of survival. It is dated Massy, May 28 1971, and he was responding to my sending him an anonymous booklet, “Mathematics for Death,” which I had written, printed, and distributed in Madison (translation of the letter follows):
« Cher Nirenberg,
Merci pour votre lettre du 10 Mai, et votre envoi de Mathematics for Death. J’ai lu ce travail avec très grand intérêt, et le fais circuler parmi les amis. Il est probable que nous aurons l’occasion d’en faire usage dans le journal Survivre. – Nous n’avons malheureusement pas de littérature sur la défoliation au Vietnam ; mais je vais demander encore autour de moi, et vous enverrai si je trouve quelque chose.
J’ai été étonné d’apprendre que vous seriez l’an prochain en Argentine. Si vous y allez à titre de savant, invité officiel et tout, vous irez comme exportateur de l’impérialisme américain, et cela me semble en contradiction flagrante avec le genre des idées représenté par Science for the People, en particulier. Il est vrai qu’il y a deux ans, je n’avais pas encore compris ce genre de choses, et j’y serais allé sans me rendre compte de rien. Mais il n’en est plus ainsi maintenant, et je pense que ce n’est rien apprendre que d’apprendre seulement dans le plan théorique, sans que cela influe immédiatement notre pratique personnelle ! Même si vous avez accepté à un moment où vous n’en voyiez pas les implications, vous pouvez refuser maintenant, en motivant publiquement votre refus … Je serais très intéressé d’avoir des détails sur vos projets.
A. Grothendieck »
[Thank you for your letter of May 10, and for your sending Mathematics for Death. I read that work with very great interest, and I circulate it among the friends. Likely we will have occasion to use it in the journal Survival. – Unfortunately we don’t have any literature on the defoliation in Vietnam, but I’ll keep asking around, and will send you whatever I find.
I was surprised to learn that you would be in Argentina next year. If you go there as a scientist, officially invited, you’ll be going as exporter of American imperialism, and that, it seems to me, is in flagrant contradiction with the sort of ideas supported by Science for the People, in particular. True, two years ago I had not yet understood this sort of thing, and I would have gone without suspecting anything. But now it is different, and I think that to learn just at a theoretical level, with no immediate change in our personal behavior, is not really learning. Even if you accepted at a time when you didn’t see the implications, you still can reject the invitation, and publicly explain why you are rejecting it. … I would be very interested in the detail of your projects.
To this I replied that I was originally from Argentina, and that I had to go for a couple of months because, among other things, my mother and my sister were still there. Grothendieck then wrote back saying that of course my case was different. But the interest of the above letter lies mainly, I think, in its style and circumstance. Notice the frequent use of sharp, unambiguous notions taken from logic and math: “If you go … (then) you’ll be going as an exporter,” “in flagrant contradiction with,” “when you didn’t see the implications,” “in particular.” —A usage not to be wondered at, since Grothendieck had spent most of his thinking life handling precisely those notions. —Yes, but here he handles them clumsily, for he applies them where they do not belong, not in a formal logical setting but in a lived one, of which furthermore he knows next to nothing: he has never inquired about my past or parentage; all he knew was that I was a mathematician and that I shared most of his political preoccupations and some of his opinions.
The confrontation of a great man with the feeling that he has been hitherto neglecting the most important thing, which he finds to be incompatible with his occupation, has all the trappings of tragedy; the case of a great mathematician who suddenly discovers the incompatibility between doing math and caring for human survival or salvation, or simply human life, throws, in addition, an intense light on our culture and our predicament. Pascal comes right to mind. He gave memorable expression to the experience of feeling the immense pride of a world-piercing intelligence together with and against the anguished consciousness of the misery and dereliction of man without God. But Pascal had the great advantage of an education in the classics (and in Montaigne) and of the theological support from his sisters and his friends of Port-Royal. Grothendieck had nothing of the sort: he was a man of our arid time, brought up in a concentration camp, quite alone; a man who lived in lofty regions—in Hegel’s words (Wissenschaft der Logik, Preface of 1831), “In the silent regions of thought which has come to itself and communes only with itself, [where] the interests which move the lives of races and individuals are hushed.” And suddenly, in or about May 1968, those moving interests were not hushed any more, indeed they were screaming through the streets at the top of their voice, they were screaming against the communion of thought only with itself, and Grothendieck took those screams to heart, and tried to listen to life: that is, I think, not less admirable than his dazzling invention of ever-subtler cohomologies, although mathematicians are likely to snigger at what I’m saying, for they often repeat that no one is free to leave math: math is a most difficult mistress who leaves her devotees when she decides. La Belle Dame sans Merci.
Grothendieck is a tragic character in more than the single sense carried by the cry, “Farewell! Othello’s occupation’s gone,” meaning that his life-long vocation was no longer possible. We could imagine Grothendieck leaving math, temporarily perhaps, and deciding to help save the world from extinction by political action, then carefully preparing himself for the new task by learning new ways of thought, new rhetorical strategies. But that was not the real Grothendieck. Even though he left math, he could not leave behind the mathematician’s sharpness of conceptual thought and the rigor mortis of logic, even though he was acutely aware of their inapplicability to what he called “everyday life”: that was, I feel, the deeper tragedy. In the public talk and dialogue of January 27 1972, titled « Allons-nous continuer la recherche scientifique ? », he had this to say:
« Les modèles que nous fournit la mathématique, y compris les modèles logiques, sont une sorte de lit de Procuste pour la réalité. Une chose toute particulière aux math, c’est que chaque chose, chaque opposition, si l’on met à part des subtilités logiques, est ou bien vrai ou bien fausse : il n’y a pas de milieu entre les deux. La dichotomie est totale. En fait, ça ne correspond absolument pas à la nature des choses. Dans la nature, dans la vie, il n’y a pas des choses qui soient absolument vraies ou absolument fausses. »
[The models provided by mathematics, including the models of logic, are a kind of procustean bed for reality. Specific to math is the fact that each thing, each opposition —save for subtle logical distinctions— is either true or false: there is no middle ground. The dichotomy is complete. In fact, that is not at all the nature of things. In nature, in life, nothing is absolutely true or absolutely false.]
Nor could he leave behind (although he tried hard) the mathematician’s arrogance, whose origin may lie in the consciousness, whether welcomed or refused, of having the keys to the basement and wine cellar of truth. I had moved to SUNY Albany in September 1971, and early in 1972 I invited Grothendieck to give a talk, which took place not at the Math Department, nor anywhere else on campus, but at the Friends Meetinghouse over on Madison Avenue. In that talk there was no question of math; only of the advisability of stopping scientific research because of the risks to humanity’s survival. I don’t think anyone in the audience was persuaded, and I know for a fact that my colleagues were not pleased. I thought Grothendieck was not advancing his cause (which was to a large extent mine) by sharply separating math questions from the pressing problems of our culture and civilization. Why didn’t he talk about math in a non-technical way, generally, historically and philosophically, and combine and motivate his political views with and against that background? After all, math, ever since the Presocratics, has been pretty much near the heart of Western thought. By refusing to have anything to do with mathematical subjects in his public talks, Grothendieck was turning off his audiences, which consisted mainly of mathematicians and scientists. He, who talked against total dichotomies, indulged in them blithely. But here’s what was hardest for me to understand: how such penetrating intelligence, so adept at dwelling in those silent regions of thought that communes only with itself, how could he think it a smart idea to use his scientific prestige as a bait for scientists to listen to him talking about the approaching apocalypse? How could he be so blind to the fact that people resented being gathered around a pulpit under false pretenses?
That evening, the boys tucked in bed and our dog with them, Isabel, Grothendieck, and I, together with Justine, the young woman Grothendieck had brought along, sat and talked in our living room. In fact Justine didn’t talk at all, but just sat. For a while Isabel and I were under the impression that she was dumb, and that, in an overflow of generosity, Grothendieck was taking care of her. Then he told us that she was a math graduate student from Rutgers, which demolished our hypothesis. Right away Isabel and I framed another: Justine was utterly intimidated in the presence of genius.
Asked about his impressions of Italy, Grothendieck replied that he had retained none, since back then (hardly two years had passed) all there was in his mind was math. Which was depressing if taken at face value (could not two sorts of beauty coexist in one soul?), but which, on second thought, pointed to a remarkable feature of the great mathematician’s psyche. I mean the way he dealt with sameness and difference. As a mathematician, his strongest impulse was to cover the greatest possible variety of structures under the same capacious tent; but he was given to cutting his own life sharply into befores and afters, eras of darkness and eras of light, and there seemed to be no higher level or standpoint from which those could be seen and felt as parts of the same I. The fact that towards the end of his life, an anchorite in the Pyrenees, Grothendieck repeatedly and most formally forbade the publication or re-publication of any of his writings, is the extreme example of his more general refusal—or inability—to consider his life as one whole. I imagine him as a legendary hero who asked of the gods the gift of seeing the unity of the world, and it was given him, at the price of never seeing the unity of his soul.
The episode that sticks most uncomfortably in my recollection of that evening has to do with Dieudonné: I may have brought him up, and elicited from our guest a disparaging word. Often Grothendieck wrote against heartless mathematicians who refer to their less talented colleagues as cons, or imbeciles, yet here he was treating his old teacher and amanuensis with cold dismissal and disdain. His letter, transcribed above, came back to my mind, and his proffered maxim: “to learn just at a theoretical level, with no immediate change in our personal behavior, is not really learning.”
Before going to bed, Grothendieck gave us precise instructions for breakfast. Justine wanted hot cereal with some fruit, and he a couple of poached eggs with white toasted bread and butter. Isabel and I were nonplussed: neither before nor after have we ever received a guest so particular. Another famous French mathematician and a founder of Bourbaki, André Weil (his daughter tells in her memoir) always insisted on having the best room everywhere he went as a guest, but Isabel and I never had the pleasure of his company. The next morning Grothendieck and his girlfriend departed, and I never saw them again.
In 1975 Survivre et vivre published its final issue. Unfortunately, because of Grothendieck’s forbiddance, lately those texts have almost entirely disappeared from the Web. The same happened with Récoltes et Semailles, a thousand-page memoir he wrote in 1985-6, and with La Clef des songes, a Renaissance-style treatise about dreams, which begins with a dream he had on April 30 1987 and which, he writes, was for him a complete renewal, a second birth. Those latter long texts should be of the greatest interest for anyone who aspires to go beyond psychology, as well as beyond logic, and study the logos of the psyche, that of which Heraclitus tells us that no matter which way we go, we will never find the end. But I am afraid most scientists and most psychologists will deal with Grothendieck, if at all, by sticking on him labels such as “bipolar disease,” “autism,” “Asperger,” or God knows what else, thereby missing the uniqueness of a soul superbly endowed for the study of objective, calculable form, yet painfully struggling to arrive at some knowledge of her non-calculable self. The nature of the difference between those two kinds of knowledge has become a mystery to us, a mystery at the heart of our civilization, just as black holes lie at the heart of galaxies.
Récoltes et Semailles is the author’s account, bitter for the most part, of his dealings with other mathematicians.
Grothendieck and Serre, 1961.
He reserves the most acerbic notes for his most brilliant student, the Belgian Pierre Deligne, and his colleague and correspondent from the early days, Jean-Pierre Serre (see photo). I am completely unable to judge on the fairness of Grothendieck’s accusations, but I will comment on just one paragraph where he speaks of Aldo Andreotti, who, you may remember, was the occasion of my meeting Grothendieck in 1970 for the first time, in Pisa:
« Aldo avait une très vive sensibilité, qui ne s’était nullement émoussée par le commerce avec la mathématique et avec des "polars" comme moi. Il y avait en lui un don de sympathie spontanée pour ceux qu’il approchait. Cela le mettait à part de tous les autres amis que j’ai connus dans le milieu mathématique, ou même en dehors. Chez lui toujours l’amitié prenait le pas sur les intérêts mathématiques communs (qui ne manquaient pas), et c’est un des rares mathématiciens avec qui j’aie tant soit peu parlé de ma vie, et lui de la sienne. Son père, comme le mien, était juif, et il avait eu à en pâtir dans l’Italie mussolinienne, comme moi dans l’Allemagne hitlérienne. » Récoltes et Semailles, p. 151.
[Aldo had a very lively sensitivity, which wasn’t at all dulled by his dealings with math and with “geeks” like me. He possessed the gift of spontaneous sympathy for those he encountered. That set him apart from all the other friends I’ve met in the math milieu, and even elsewhere. With him, friendship always came before our mathematical common interests (which were not lacking); he’s one of the very few mathematicians with whom I have talked, if ever so little, about my life, and he about his. His father was Jewish like mine and he suffered on that account in Mussolini’s Italy, like me in Hitler’s Germany.]
Nothing in Récoltes et Semailles surprised me more than this short paragraph, no doubt because I had got to known Andreotti fairly well that year I spent in Pisa, 1969-70. Grothendieck’s image of the man and mine have few significant elements in common. In my recollection, the young mathematicians in Pisa did not see Andreotti as a fountain of spontaneous sympathy, but then of course no man behaves in the same way toward everyone, and Andreotti was no exception to that rule. He and Grothendieck were not lacking in common mathematical interests: at the time, Andreotti was working on cohomology problems, and could hope to get some help from his guest, the master in the subject. That Aldo Andreotti’s father, the Tuscan sculptor Libero Andreotti (died 1933, Florence) was a Jew I find hard to believe. If he was, it must have been a very well kept secret; Andreotti is not an Italian-Jewish name, and no Jew would willingly be buried at the cemetery delle Porte Sante in the church of San Miniato al Monte in Florence. The encyclopedic Dizionario Biografico degli Italiani (Enciclopedie Treccani, 1961) contains no hint that Libero Andreotti might have been a Jew: it cites four public homages to him between 1938 and 1945, the period of official persecution of the Jews in Italy; one of them, dated 1943, is by Carlo Carrà, who in those days was one of the most celebrated Italian artists supporting and glorifying the regime.
“Libero the Jew” may have been a result of Grothendieck’s misunderstanding, but I find that the Jewish Daily Forward picked it up:
“After the war, Grothendieck formed close friendships with fellow mathematicians, especially Aldo Andreotti, (1924-1980) an Italian specialist in algebraic geometry who ‘suffered in Mussolini’s Italy as [Grothendieck] did in Hitler’s Germany,’ he explained, because Andreotti’s father was Jewish.” (Jewish Daily Forward, note by Benjamin Ivry.)
Aldo himself and his wife Barbara were devout Catholics; a priest went periodically to their house to bless their rooms. It was a time when Italian society and politics were sharply polarized, and one sure road to academic advancement was adherence to either the Communist Party or the Catholic Church; the more bizarre, then, that a spirit forged on the anvil of anarchism and libertarianism, like Grothendieck’s, would claim to have found its mate in an ecclesiastical one nourished at the altar. In conclusion, by the time Grothendieck finished Récoltes et Semailles in 1986, his inability to understand people and to gauge their intentions was as complete as fifteen years earlier, when he was exhorting me not to export American imperialism to Argentina. He withdrew, first into the contemplation of his dreams, and finally, around 1990, into total seclusion in the village of Lasserre, in the Ariège.
The following article from the local newspaper was sent to me by Roy Lisker, who knew Grothendieck and who, I understand, is translating Récoltes et Semailles into English; it gives us some idea of how somber was the tragic denouement:
I find no better adieu to that great, tormented soul than the following lines from Matthew Arnold’s “The Scholar-Gipsy”:
“Fly hence, our contact fear!
Still fly, plunge deeper in the bowering wood!
Averse, as Dido did with gesture stern
From her false friend's approach in Hades turn,
Wave us away, and keep thy solitude!”