收获与播种

Presentation of the Themes or PRELUDE IN FOUR PARTS — p.9
1. By way of a foreword — p.9
2. A walk through a life's work, or the child and the Mother — p.13
  2.1 The magic of things — p.13
  2.2 The importance of being alone — p.16
  2.3 The inner journey - or myth and testimony — p.18
  2.4 The painting of mores — p.20
  2.5 The heirs and the builders — p.21
  2.6 Viewpoint and vision — p.24
  2.7 The "great idea" - or the trees and the forest — p.26
  2.8 The vision - or twelve themes for a harmony — p.28
  2.9 Form and structure - or the way of things — p.32
  2.10 The novel geometry - or the marriage of number and size — p.34
  2.11 The magical fan - or the innocence — p.36
  2.12 Topology - or surveying through the mist — p.38
  2.13 Topoi - or the double bed — p.41
  2.14 Mutation of the notion of space - or breath and faith — p.43
  2.15 All of the king's horses... — p.45
  2.16 Motives - or the heart within the heart — p.46
  2.17 Toward the discovery of the Mother - or the two sides — p.50
  2.18 The child and the Mother — p.53
  Epilogue: the invisible Circles — p.54
  2.19 Death is my cradle (or three toddlers for one moribund) — p.54
  2.20 A look at the next-door neighbors — p.56
  2.21 "The unique" - or the gift of solitude — p.60
3. A letter — p.65
  3.1 The one-thousand page letter — p.65
  3.2 Birth of Récoltes et Semailles (a lightning fast retrospective) — p.66
  3.3 The death of the boss - or abandoned construction sites — p.68
  3.4 A wind of burial... — p.72
  3.5 The journey — p.76
  3.6 The obscure side - or creation and disdain — p.78
  3.7 Respect and fortitude — p.81
  3.8 "My close acquaintances" - or connivance — p.83
  3.9 The plunder — p.87
  3.10 Four waves in one movement — p.89
  3.11 Movement and structure — p.93
  3.12 Spontaneity and rigor — p.96
  Epilogue in postscript - or context and preliminaries to the debate — p.98
  3.13 The bottle spectrograph — p.98
  3.14 Three feet in a single dish — p.99
  3.15 Gangrene - or the spirit of our times (1) — p.103
  3.16 Honorable amend - or the spirit of our times (2) — p.105
I First part: Fatuity and Renewal — p.109
II Second part: The Burial (I) - or the Robe of the Emperor of China — p.111
III Third part: The Burial (II) - or the Key to the Yin and the Yang — p.193
IV Fourth part: The Burial (III) - or the Four Operas — p.201

All that was left to write was the foreword, in order for Récoltes et Semailles to be given to the publisher. And I swear that I went into it with all the will in the world to write something suitable. Something reasonable this time. No more than three or four pages, carefully phrased, that would introduce this huge tome of more than one thousand pages. Something which "grabs" the attention of the jaded reader, which perhaps suggests that in these frightening "more than a thousand pages", there could be things of interest to him (or things which concern him, who knows?). It is not really my style to pander. But I was ready to make an exception for once! The "publisher crazy enough to give it a shot" (to publish this visibly unpublishable monster) had to make ends meet one way or another.

But then, it didn't come. And yet I tried my best. And not only for an afternoon, as I originally planned. Tomorrow will mark three weeks since I started, since the sheets began accumulating. What came, for sure, isn't what one could decently call a "foreword". It is yet another miss! Blame it on my old age - I have never been a salesman. Even when it comes to pleasing (oneself or friends...).

What came instead is a sort of long "Walk" with commentary, through my work as a mathematician. A Walk intended mostly for the "layman" - he who "never understood anything about math". And for myself as well, having never indulged in such a Walk. Step by step, I found myself unearthing and saying things that had previously remained unspoken. As if by chance, these are also things which I feel are most essential, both in my practice and its outcome. They are things which are not technical in nature.

It will be up to you to decide whether or not I succeeded in my naive enterprise to "get the message through" - an enterprise which surely is also a bit mad. My satisfaction and my pleasure will come from making you feel these things. Things that many of my wise colleagues do not feel anymore. Maybe they have become too wise and too prestigious. This often leads to losing touch with the simplest and most essential things.

During this "Walk through a body of work" I also speak of my life. As well as, here and there, what Récoltes et Semailles is about. I mention this in more detail in the "Letter" (dated from May of last year) which follows the "Walk". This Letter was directed towards my previous students and to my "old friends" in the mathematical community. But even the Letter is not technical in nature. It can be read by any reader interested in learning through a "heartfelt" narrative, the odds and ends that led me to writing Récoltes et Semailles. Even more than the Walk, the Letter will provide a preview to a particular atmosphere in the "prestigious mathematical world". And also (just as in the Walk) of my writing style, as peculiar as it may seem, and of the spirit that is expressed through this style - a spirit which is not universally appreciated.

In the Walk and throughout Récoltes et Semailles, I speak of the activity of doing mathematics. It is an activity for which I have first-hand experience and know very well. Most of the things I say about it can surely be said of any kind of creative work, or work involving discovery. In any case it is true of all "intellectual" work, that which is done using the "brain" and in writing. All such work proceeds through the outbreak and development of an understanding of the things which are being probed. But to take an example at the opposite extreme, romantic passion is also an activity of discovery. It opens us to understanding of a "physical" nature which also renews itself, develops, and deepens over time. Both of these impulses - that which, say, livens the mathematician at work and that of the lover - are much closer in nature than we generally assume or we readily admit. I hope that the pages of Récoltes et Semailles will make you feel this impulse in your work and in your daily life.

Most of the Walk focuses on mathematical work itself. I remain mostly silent concerning the context in which this work takes place, and concerning the motivations at play outside of mathematical work itself. This risks giving me, or the mathematician, or the "scientist" in general a flattering but deformed image. In the style of "grand and noble passion" without any form of rectification. In accordance with the great "Myth" of Science (with a capital S, if you will!). The heroic myth, "promethean", to which writers and thinkers have succumbed (and continue to succumb). Only historians, maybe, manage to sometimes resist this tantalizing myth.

The truth is, within the motivation of these "scientists", which sometimes lead them to devote themselves entirely to their work, ambition and vanity play a role just as important and universal as they do in any other profession. This phenomenon appears in blunt or subtle ways depending on the person - and I am no exception to this pattern. The reading of my testimony will hopefully leave no doubt about this fact.

It is true also that even the most intense ambitions are powerless at discovering or proving a novel mathematical statement - just as they are powerless (for instance) to "make one hard" (in the proper sense of the term). Whether man or woman, what "makes one hard" is not ambition, nor the desire to shine, to exhibit power, of a sexual nature in this case - quite the contrary! It is the acute perception of something strong, at once very real and very delicate. One could call it "beauty", though this is one of a thousand faces of this thing. Being ambitious doesn't prevent one from sensing the beauty of a being or a thing. But it is not ambition which makes us feel it...

The person that first discovered and mastered fire was somebody just like you and me. Not at all what we refer to as "hero", or "demi-god", and so on. Surely, just like you and me, he has encountered the grip of anxiety as well as the time-worn remedy of vanity which alleviates the grip. At the instant at which he "knew" fire, there was no fear nor vanity. Such is the truth in the heroic myth. The myth becomes insipid when it is used to disguise another aspect of things which is just as real and essential.

My aim in Récoltes et Semailles has been to address both aspects of the myth - that of the impulse towards understanding, and that of fear and its vain antidotes. I believe I "understand", or at least know this impulse and its origin (or perhaps one day I will discover to what extent I was deluded). But concerning fear and vanity, as well as the resulting insidious creativity blocks, I know that I have yet to thoroughly uncover this great enigma. And who knows if I will ever reach the conclusion of this mystery in the year I have left...

As I was writing Récoltes et Semailles two images emerged in order to represent the two aspects of the human journey: that of the child (aka the worker), and that of the boss. In the Work which we are about to undertake, we will be dealing mostly with the "child". It is him also that is featured in the subtitle "The Child and the Mother". The motivation for this name will hopefully become clear over the course of this work.

In the remainder of this reflection, however, it is the Boss who takes the lead. He is living up to his name! It would be more accurate to speak of multiple bosses of competing enterprises rather than of a singular boss. But it is also true that all bosses essentially resemble one another. And once we mention bosses it is implied that we will also have to deal with "villains". In part I of the reflection (named "Fatuity and Renewal", which follows the present introductory section). I mostly take on the role of the "villain". In the following three parts it is mostly the "others". Chacun son Tour! That is to say that, in addition to philosophical reflections and "confessions" (not contrite), there will be "vitriolic portraits" (to use the expression of one of my colleagues who found himself tormented). Not to mention large-scale, well-oiled "operations".

Robert Jaulin assured me (half jokingly) that in Récoltes et Semailles I was making the "ethnology of the mathematical community" (or maybe the sociology I do not quite remember). It is flattering of course to learn that one has been (unknowingly) doing scholarly things! It is true that during the "investigation" segment of the reflection, I saw in passing, in the pages I was writing, a good chunk of the mathematical establishment without counting a number of my colleagues and friends of more modest status.

Over the past few months, since I have been sending preliminary versions of Récoltes et Semailles this has been "brought up" again. My testimony arrived like a tome landing in a pond. There were responses of every kind (except for boredom...). Yet almost every time the response was far from what I expected. There was also a lot of silence, which speaks volumes. Visibly, I had (and still have) a lot to learn about what happens in people's minds, among my previous students and other colleagues - excuse me I meant about "the sociology of the mathematical milieu"! To all those that contributed to the great sociological work of my old days, I would like to express sincere recognition.

Of course, I was particularly sensitive to warm responses. There were also some rare colleagues who conveyed a sentiment (thus far unexpressed) of crisis, or of degradation of the inner workings of the mathematical milieu with which they identify themselves.

Outside of this milieu, among the very first to respond positively to my testimony, I would like to recognize Sylvie et Catherine Chevalley, Robert Jaulin, Stéphane Deligeorge, Christian Bourgois. If Récoltes et Semailles achieves a wider diffusion than that of the initial printing (addressed to a very limited social circle of people), it is mostly thanks to them. Thanks mostly to them communicating their conviction that what I strived to seize and say had to be said. And that it could have an audience outside of my colleagues (who are often sullen, sometimes even belligerent, and strictly opposed to question their position...). Indeed Christian Bourgois did not hesitate to risk publishing the unpublishable, and Stéphane Deligeorge did not hesitate to place my indigestible testimony alongside works of Newton, Cuivier, and Arago (I could not ask for better company). To each of them, for their repeated expressions of sympathy and trust, intervening at an especially sensitive moment, I happily extend all my gratitude.

And here we are at the beginning of a Walk through a life's work, serving as a prelude to a journey through a lifetime. A long journey, over a thousand pages long, each of which is densely packed. I spent a lifetime undergoing this journey without ever exhausting it, and it then took me more than a year to rediscover it, one page at a time. Words were sometimes hard to come by, as they were intended to convey an experience which evaded comprehension - just as ripe grapes stacked in a press occasionally seem to evade the force upon them... But even in those moments when words come flooding, it is not by happenstance. Each word has been carefully weighed in passing, or after the fact.

Thus, this reflection/testimony/journey is not meant to be read hastily, in a day or in a month, by a reader rushed to reach the final word. There is no "final word", no "conclusion" in Récoltes et Semailles, no more than there are any such things in my life or in yours. There is only a wine, aged over the course of a lifetime, at the core of my being. The last glass which you will be drinking will be no better nor worse than the first or the hundredth. They are all "the same", and they are all different. And if the first glass is spoiled, so is the rest of the barrel; it is better to drink fresh water (if such can be found), than to drink bad wine. But a good wine ought not to be drunk in haste, nor expeditiously.

When I was little I liked going to school. The same teacher taught us reading, writing, arithmetic, singing (he accompanied us with a small violin), and he even told us about prehistoric men and the discovery of fire. I do not ever recall being bored at school during those days. There was the magic of numbers, that of words, of signs, and of sounds. That of rhymes as well, through songs and small poems. There seemed to be, within rhymes, a mystery which extended beyond words. I believed this until the day somebody told me that there was a simple "trick" to it; that rhyme was simply when one ends two consecutive spoken movements by the same syllable, so that, as if by magic, these phrases became verses. It was a revelation! At home, where I found a good audience, for weeks or months on end, I amused myself by making verses. At one point, I even started exclusively speaking in rhymes. That period has passed, fortunately. Yet even to this day, I still sometimes write poems - but without trying to force the rhyme, if it doesn't seem to come by itself.

On another occasion an older friend, who was already in high school taught me about negative numbers. It was also a fun game, although I rapidly exhausted it. And then there were crosswords - I spent days and weeks constructing them, ever more interwoven. Within that game the magic of form, that of signs, and that of words found themselves combined. But even that passion subsided without leaving a trace.

During my high school years, the first of which took place in Germany and the others in France, I was a good student without quite being the "star student". I devoted myself without restraint to the courses which I cared most about, and tended to neglect the others, without really caring for the impression I made on my "prof". During my first year of high school in France, in 1940, I was interned at a concentration camp with my mother at Rieucros, near Mende. It was wartime, and we were foreigners - "undesirables", as they said. But the administration of the camp turned a blind eye towards the kids, however undesirable they may be. We came and went as we pleased. I was the oldest, and the only one to go to high school. It was a four or five kilometers long walk, often in rainy and windy weather, wearing makeshift shoes that always got wet.

I still remember my first "math examination", in which the teacher gave me a bad grade, for my proof of one of the "three cases of equality of triangles". My proof wasn't exactly that of the book, which he followed religiously. Yet, I knew very well that my proof was no less convincing than that of the book which I followed in spirit, through repeated invocation of the traditional "we slide this figure in such and such a way onto that figure". Visibly, this teacher did not feel capable of judging things (namely, the validity of my reasoning) on his own. He had to report to a higher authority, that of a book in this case. Since I still remember this incident, I must have been stricken by such dispositions. In the years that followed, I have been presented with more than enough evidence to realize that such dispositions are far from exceptional - rather, they are the quasi-universal norm. There is a lot to be said on this subject - one which I approach more than once from various angles in Récoltes et Semailles. Somehow, to this day, I find myself invariably taken aback whenever I am confronted with such behavior...

During the last few years of the war, while my mother was still interned at the camp, I lived in a youth refugee house called "Secours Suisse", in Chambon sur Lignon. Most of us were jewish, and when we were told (by the local police) that there would be raids by the Gestapo, we went to hide in the woods for a night or two, in small groups of no less than 3, without quite realizing that our life was on the line. The region was filled with jews hiding in Cévenol country, and many of us survived thanks to the solidarity of the local population.

What struck me most about "Collège Cévenol" (where I was raised), was the extent to which my peers were disinterested in learning. As for myself, I devoured our textbooks at the beginning of the school year, thinking that this time around, we would finally learn truly interesting things; and for the rest of the year I utilized my time as best as I could while classes dragged along inexorably one trimester at a time. Yet we had some wonderful professors. The natural history professor teacher, mister Friedel, was a person with remarkable intellectual and social qualities. However, as he lacked authority, the class was acting out of control, to the extent that it became impossible to hear what he had to say, as his voice was lost in the hurly-burly. This may be the reason I haven't become a biologist!

I spent a fair amount of time, including class time (shh...), solving math problems. The ones in the book soon became insufficient. Perhaps it was because they tended to resemble one another after a while; but mostly, I believe, because they seemed to come out of the blue à la queue-leue-leue, with no indication as to where they came from or where they're going. These were the books problems, not mine. And yet, natural questions were plentiful. For instance, once the three side lengths a, b, and c of a triangle are known, so that the triangle itself is known (up to its position), there has to be an explicit "formula" that expresses the area of the triangle as a function of a, b, and c. Likewise, for a tetrahedron of which the six side lengths are known - what is the volume? I struggled through that one for a bit, but I must have gotten there eventually. In any case, when a problem "grabbed me", I did not count the hours or days that I spent working on it, even if it meant losing track of everything else! (And such remains the case to this day...)

What I found least satisfying, in our math textbooks, was the absence of a serious definition of the notion of length (of a curve), of area (of a surface), or of volume (of a solid). I promised myself to make up for this omission as soon as I could. This is what I devoted most of energy to between the years of 1945-1948, while I was a student at the University of Montpellier. University lectures weren't for me. Without ever quite realizing it, I must have been under the impression that all my professor did was recite the contents of the textbooks, just like my first math teacher at the lycée de Mende. I barely ever set foot on university grounds, just enough to keep up to date with the perennial "program". Books sufficed to cover said program, but it was clear that they offered no answers to the questions I was asking myself.

Truly, they did not even see them, no more than my high-school textbooks did. As long as we were provided with recipes for all sorts of calculations, such as lengths, areas, volumes, through single, double, triple integrals (dimensions higher than 3 were carefully avoided...), the problem of providing an intrinsic definition was omitted by both my professors and textbook authors.

From my then limited experience, it seemed that I was the only person in the world to be gifted with a curiosity for mathematical questions. Such was, in any case, my unexpressed conviction, during those years spent in complete intellectual solitude (which did not bother me). To be fair, it never occurred to me at that time to investigate whether or not I was the only person in the world to take interest in what I was doing. My energy was sufficiently absorbed by the task I set for myself: to develop a fully satisfactory theory.

I never doubted that I would succeed in reaching the end of the story, as long as I was committed to scrutinizing these structures, spelling out on paper all the intuitions and insights that would come to me, one by one...

The text which I am hereby sending you, of which a limited number of copies were typed and printed by my university, is neither an off-print, nor a preprint. Its title, Récoltes et Semailles, makes this much clear. I am sending it to you the way I would send a long letter - including the personal dimension, indeed.

If I have decided to send it to you, rather than await for you to learn about it some day (if your curiosity leads you to it) in the form of some publicly available volume in a library (if there even exists an editor crazy enough to engage in such an adventure...), it is because I am addressing this letter to you more than to others. I have thought of you more than once in the course of writing it - I must say that I have now been writing this letter for more than a year, and devoting all my energy to the task. It is a gift I am making you, and I took great care in the process to give out what I had best to offer (at any given moment).

At the same time as I am sending you Récoltes et Semailles, I am also sending it to all of the colleagues, friends, and (ex-)students of the mathematical world with whom I was close at one time or another, as well as to those who appear in my reflection in some form, both named and unnamed. There is a chance that you yourself appear in what follows, and if you make the effort to read it not only with your eyes and head but also with your heart, you will surely recognize yourself even in places where you are not explicitly named.

Récoltes et Semailles contains many things, and many will see them as several different things: a voyage of discovery through the past; a meditation on existence; a painting of mores of a milieu and an era (or a painting of the insidious and unstoppable transition from one era to the next...); an investigation (almost police-like at times, and elsewhere approaching the style of a cloak-and-dagger novel taking place in the underbelly of the mathematical megapolis...); a vast mathematical divagation (which will lose more than one reader...); a practical treaty in applied psychoanalysis (or, alternatively, a book of "psychoanalytic fiction"); a eulogy of self-knowledge; "My confessions"; a private diary; a psychological study of the processes of discovery and creation; an indictment (unforgiving, as it must...), or even a settling of accounts in the mathematical "beau monde" (in which no punches are pulled...).

If there is one thing I can guarantee, it is that I never once was bored in the process of writing it, and that from it I have learned and seen a great deal. If you find the time to read it among your other important duties, I doubt that you will be bored. Unless you force yourself to read it, who knows...

As you know, I left the "grand monde" of mathematics in the year 1970, following an issue of military funding at my home institution (the IHES). After a few years devoted to anti-military and ecological activism, in the style of a "cultural revolution" - about which you probably received echoes here and there - I practically disappeared from the public sphere, and settled at a remote provincial university. Rumor has it that I am spending my days herding sheep and digging wells. The truth is that, in addition to several other occupations, I am bravely continuing to give university lecture, like everybody else (teaching was my first source of income, and it remains so to this day).

In 1976, a new passion appeared in my life, one about which I felt as strongly as I once felt about mathematics, and which is closely related to the latter. It was a passion for what I call "meditation" (things need to be named after all). This name, like any other, risks leading to countless misunderstandings. As for mathematics, the process of meditation is one of discovery. I express myself on this subject at various points in Récoltes et Semailles. It visibly held enough in store to keep me occupied for the rest of my life. In fact, I have more than once gotten to thinking that mathematics was now a matter of the past, and that it was time for me to orient myself towards more serious matters - time to "meditate".

This project began two years ago, during the spring of 1983. I was then already too busy writing (volume 1 of) "Pursuing Stacks" ("À la Poursuite des Champs" in French), which also was to constitute volume 1 of the (mathematical) "Reflections", to stop and reflect on what was happening to me. Nine months later, as is fit, this first volume was virtually complete, and all that was left for me to do was to write the introduction, re-read everything, add annotations - and off it could be sent to be printed...

The volume in question is still not finished to this day - it hasn't moved by an inch for the past year and a half. The introduction that I had left to write grew to take over twelve hundred pages (typewritten), and when all will be said and done I estimate it will be some fourteen hundred pages long. You will have guessed that this "introduction" is nothing but Récoltes et Semailles.

During these fifteen years of intense mathematical work, a vast unifying vision had hatched, matured, and grown within me, taking the form of a handful of very simple idées-forces. It was the vision of an "arithmetic geometry", a synthesis of topology, geometry (both algebraic and analytic), and arithmetic, a first embryo of which I had found in the Weil conjectures. This vision was my principal source of inspiration during those years, a period which I mostly remember as the one during which I managed to formulate the key ideas of this novel geometry, and to develop some of its main tools.

Over time, this vision and these "idées-forces" became second nature to me. (And this feeling of "second nature" persists in me to this day, despite having ceased all contact with these ideas for nearly fifteen years!) I found them to be so simple, so obvious, that it was natural for "everyone else" to internalize them and to make them their own over time, at the same time as I went through these motions myself. It is only recently, during the past few months, that I realized that neither this vision, nor the "idées-forces" which had been my constant guides, could be found in writing in any existing publication, save perhaps for tacit appearances between the lines.

Most importantly, I also realized that this vision which I thought I had imparted to others, and the "idées forces" which carry it, remained ignored by all to this day, twenty years after having reached their full maturity. I alone, the worker and servant to these things which I had the privilege of discovering, remain the sole vessel in which they have remained alive.

Some tool or other which I have crafted will find itself used in various places to "break open" a problem reputed for being difficult, the way one would break open a safe. The tool is visibly solid. Yet, I am aware of a "force" it has other than that of a crowbar. The tool is part of a Whole, just as a limb is part of a body - it is part of a Whole from which it is issued, which gives it its full meaning and infuses it with strength and life energy.

Upon his departure, it was as if a masonry enterprise had gone bankrupt, following the unexpected passing of the boss: from one day to the next, so to speak, the construction sites were deserted. The "workers" were gone, each of them carrying some small gewgaw which they thought may be of use at a later time. The cash register was gone, and as such there was no longer any reason for them to tire themselves out at work...

I couldn't help but to feel sorrow over the fact that everything seemed to have stopped in its tracks; I no longer heard news about motives, topoi, the six functor formalism, De Rham and Hodge coefficients, nor about the "mysterious functor" which was supposed to unite under one umbrella De Rham and l-adic coefficients for all prime numbers, nor about crystals (except to learn that they remained at a standstill), nor about the "standard conjectures" and other conjectures which I had formulated and which, evidently, represented crucial questions.

Even though I chose to ignore the diffuse perception of a large scale Burial for years on end, this burial nonetheless obstinately returned to haunt me in different guises, less innocuous than that of a mere disaffection with my work. I slowly came to realize, in ways I could not quite explain, that many of the constituent notions of the forgotten vision had not only fallen into disuse, they had also become, within a certain "beau monde", the objects of condescending disdain. Such was notably the case for the crucial and unifying notion of topos, which lays at the very heart of the novel geometry - and which provides a common geometric intuition for topology, algebraic geometry, and arithmetic.

In fact, it was my very name which, insidiously, mysteriously and over the course of the years, had become an object of derision - becoming a synonym for rambling discourses ad infinitum (such as those I produced on the famous "topoi", or on these "motives" that I kept dwelling on and which nobody had ever seen...), for counting angels dancing on the head of a pin for thousands of pages on ends, and for plethoric and gigantic discussions about things which everyone already knew anyway without having to read about them somewhere...

Truth be told, these "whiffs of subtle derision" which reached me here and there did not affect me that much. They remained in a way anonymous, up until three or four years ago. I saw in them a sign of somewhat bleak times, but they did not appear to be directly targeting me, eliciting neither anxiety nor concern. What did affect me more directly were signs of a distancing away from my person which I received from several of my old friends in the mathematical world, friends to whom I continued (my departure from a mutual world notwithstanding) to feel connected through sympathy, in addition to the links created by a shared passion and a common past.

The episode which felt like a "full-on punch in the face" (of someone else) took place in October 1981. This time around, and for the first time since frequent signs of a new spirit had started reaching me, I was affected - certainly more so than I would have been had I been the one who was punched, rather than someone else whom I held in affection. This someone else was a student of mine, as well as a remarkably gifted mathematician, who had just accomplished beautiful things - but that was just a detail. What wasn't a detail, on the other hand, was that three of my students from "before" showed direct solidarity in favor of an act which the person in question received (rightly) as a humiliation and an affront.

This passage "clicked" for the friend who read the previous section "the weight of a past". He wrote: "for many of your old students, the aspect, as you put it, of an invasive and borderline destructive "boss" remains strong. Whence the impression you hold."

It is true that I was looking first and foremost, in the students that approached me, for collaborators with whom to develop intuitions and ideas which had already formed within me, to "push along", in sum, a carriage that was already there, which they did not have to summon from some kind of void, "something which my correspondent had to do". This summoning - the act of bringing into being a tangible, supple, intense body of work from the intangible mist - had indeed always been, for me, the most fascinating aspect of mathematical work, as well as the part in which I most strongly felt a process of "creation" the "spirit of something more delicate and essential than a mere result".

Thus, it seems to me that the failure that I am today confronting rests on subtler causes than the kind of themes which I suggested, or the extent to which said themes remained nebulous or were clearly delineated. My role in this failure seems due rather to attitudes of fatuity within me, in the way I interacted with mathematics; attitudes which I have examined in the course of this reflection. These attitudes were bound to more or less strongly influence, if not the work itself with a given student, at least the atmosphere surrounding my person.

I would like to take the time to say a few words concerning the mathematical notions and ideas, among those which I have brought to life, which seem (by far) to be the farthest reaching. I will be mostly speaking about five closely linked key-notions, which I will briefly review in increasing order of specificity, richness, and depth.

The first idea in question is that of derived categories in homological algebra (cf. note 48 p. 274), and of their use as a "catch-all" formalism called the "six operations formalism" (namely ⊗L, Lf∗, Rf!, RHom, Rf∗, Lf!) on the cohomology of the most important kinds of "spaces" introduced to this day in geometry: "algebraic" spaces (such as schemes, schematic multiplicities, etc...), analytic spaces, topological spaces.

The progressive discovery of this duality formalism and of its ubiquitousness happened through a solitary, persistent, and exacting reflection which took place between the years 1956 and 1963. It was during the course of this reflection that the notion of derived category slowly appeared, and with it an understanding of the role which it played in homological algebra.

For almost 10 years, in the absence of the encouragement of those among my old students who were best positioned to offer it, and to support him through their interest and their experience which they had gained through their work with me. Zoghman Mebkhout produced his remarkable work in a near total state of isolation. This did not prevent him from discovering and proving two key theorems in the context of a new crystalline theory which was slowly coming into being in the midst of a general indifference.

I now come to the second pair of notions, namely that of schemes and the tightly related notion of topoi. The latter is a more intrinsic version of the notion of site, which I introduced in order to formalize the topological intuition of "localization". I was led to introduce the notions of scheme and topos one after another in response to the glaring needs of algebraic geometry. This pair of concepts carried within them the potential for a profound renewal of both algebraic and arithmetic geometry and of topology, through a synthesis of these "worlds", kept apart for too long, within a common geometric intuition.

On the other hand, this vast synthesis that would also encompass topology is still biding its time, even though the essential ideas and principal technical tools appear to have been in place for twenty years. During the fifteen years that followed my departure from the world of mathematics, the fertile unifying idea and powerful tool for discovery that is the notion of topoi has been maintained by some customary decree outside of the range of notions deemed serious. To this day, few topologists are even aware of the existence of this potentially considerable enlargement of their science, and of the novel resources which it offers.

The fifth notion which is close to my heart, perhaps more than any other, is that of "motives". It is distinct from the preceding four ideas in that "the" correct notion of motive (be it only over a base field, without even mentioning the case of an arbitrary base scheme) has not been given a satisfactory definition to this day, even if we are to accept all "reasonable" conjectures which one may need to this end.

It is just as clear today for motives as it once was for the infinitesimally small, that such beasts exist, and that they manifest themselves in every corner of algebraic geometry, as long as one is interested in the cohomology of algebraic varieties and families of such varieties, and more specifically in the "arithmetic" properties of such objects.

Nonetheless, the few people in the know (and who follow the trends) know that beginning with the premises, which remain secret, one can prove many things. It should be said that as of today, in fact since the notion appeared in the wake of the Weil conjectures (proven by Deligne, which proves a point I guess!), the yoga of motives very much exists. But it has the status of a secret science with very few initiates.

As is well known, the theory of derived categories is due to J. L. Verdier. Before he undertook the foundational work which I suggested to him, I had confined myself to working with derived categories in a heuristic way, using a provisory definition (which later turned out to be the right one) and an equally provisory intuition about the essential internal structure.

Perhaps I am the only one to regret that neither the introductory text not the foundations per se have been published, so that the essential technical prerequisites to using the language of derived categories find themselves scattered across three different sources in the literature. This absence of a systematic reference of a caliber comparable to Cartan-Eilenberg's classical book appears to me as both a cause and a sign, typical of the disaffection which took hold of the formalism of derived categories following my departure from the mathematical world in 1970.

Yet, regarding the first of the above themes, all of the essential ideas needed for a large scale foundational work seemed to have been established during the years leading to my departure. (Including the key idea of "derivator", or "machine producing derived categories", which seems to be the richest object in common to the triangulated categories encountered to this day; this idea was eventually developed to some extent in the non-additive context, nearly twenty years later, in a chapter of volume 2 of Pursuing Stacks.)

It is true that the disaffection for the very notion of derived category during the past fifteen years, which for some was connected to a disavowal of a time past, is in line with a certain fashion which suggests that we look down upon any reflection touching to foundations, however urgent it may be. On the other hand, it is also clear to me that the development of etale cohomology, which "everybody" uses nowadays without thinking twice (if only implicitly via the now proven Weil conjectures...) would not have been possible without the conceptual background consisting of derived categories, the six operation formalism, and the language of sites and topoi.

Remarkably, in reading this first publication which marks (twelve years after my departure from the mathematical world) a modest re-entry of the notion of motive into the apparatus of admissible mathematical notions, I could find nothing which would indicate to the uninformed reader that my humble person was involved in any way in the origins of this notion, long considered a taboo. Nor is there any allusion to an authorship of some form behind the development of a rich and precise "yoga", which appears in the article (in piecemeal form) as if it came out of the void.

When, just three weeks ago, I laid down my thoughts on the yoga of motives in a page or two, qualifying it as one of the "orphans" whom I held closer to my heart than any other, I must have been sorely mistaken! Surely was I dreaming, when I seemed to remember years of gestation of a vision, tenuous and elusive at first, and growing richer and more precise over the course of the months and the years, as a result of a persistent effort to grasp the common "motive", the common quintessence of which the several cohomology theories known at the time were but various incarnations.

The simple act of writing, naming, and of describing - if only to describe elusive intuitions or mere "hunches" reticent about taking concrete form - possesses a creative power. There lies the most important instrument in enacting the passion for knowledge, when the latter is invested into things which can be apprehended by the intellect. In the process of discovery for such matters, this work is the most creative step of all; it always precedes proof and enables it - or rather, without it, the question of "proving" something would not even arise, as nothing pertaining to the heart of the matter would have yet been formulated and seen.

The crude scam (which doesn't seem to inconvenience anybody, since it comes from such a prestigious character) consists in outright obfuscating the filiation of a novel and profound notion - that of motive - as well as the rich web of intuitions which I have developed surrounding this notion, under the derisory pretext that the technical approach taken to study this notion (via absolute Hodge cycles instead of algebraic cycles) is (maybe, if the Hodge conjecture is false) different from the one which I had (provisionally) adopted.

I remember (even though I have forgotten so many things!) the ever-renewed pleasure which I took in discussing with the person who had quickly become my confidant on everything which captivated me, as well as each step forward and enchanting discovering in my day-to-day love story with mathematics - and as such he never really was a "student". His perennial receptiveness and the ease with which he learned about each thing ("as if he had always known it...") acted as a constant source of enchantment. He was an ideal listener, moved by the same thirst for understanding as my own - he has an extremely sharp ear, signaling a communion between us.

Ever since we first met, I had felt that his "abilities", as we say, were of a very rare quality, and far exceeded the modest abilities which I myself possessed, even though we were of the same breed when it came to our passion for understanding and our exigency regarding the comprehension of mathematical things. I also sensed, dimly, without yet being able to put my finger on it that this "strength" which I noticed in him (and which I also noticed in myself, although present to a lesser degree) of "seeing" the obvious things which nobody else could see was the faculty of a child as well as the innocence of a child's eyes.

The blossoming I had in mind was to be the fruit of innocence, hungry for knowledge, and always ready to rejoice at the beauty of things both small and large in this inexhaustible world of ours, or in some particular region of this world (such as the vast universe of mathematical things...). Only a blossoming of this kind holds the potential for a profound renewal, being the renewal of oneself or that of one's understanding of things in the world. This potential was fully realized, I believe, in the humble man Riemann. This true blossoming removes one from disdain; be it disdain towards others (whom one feels are far below oneself...) or disdain towards things that are too "small" or too obvious to warrant attention, or which are deemed below one's legitimate expectations; or even disdain for a given dream, which is insistently communicating things which we claim to love... Disdain thereby becomes as foreign as the self-satisfaction which feeds it.

However, the student cannot surpass the master while simultaneously inwardly disavowing him, secretly laboring towards the suppression of any trace of what he had contributed (whether that contribution had been for the better or the worse...), both from his own eyes and the eyes of others - in the same way that the son cannot truly surpass the father while simultaneously disavowing him. This is something that I have learned mostly through my relationship with my own children, but also (later) through my relationships with certain past students of mine; mostly the one student, among others, whom I had always refrained from calling a "student", having felt since the day we met that I was about to learn from him as much as he would be learning from me.

The Iniquity or a feeling of return 75 · The prestidigitator 75'' · Perversity 76 · Thumbs up! 77 · The robe of the emperor of China 77' · Encounters from beyond the grave 78 · The victim - or the two silences 78' · The Boss 78'' · My friends 79 · The tome and high society - or moon and green cheese... 80

Thesis on credit and risk-proof insurance 81 · The right references 82 · The jest, or the "weight complexes" 83

Silence 84 · Solidarity 85 · Mystification 85' · The defunct 86 · The massacre 87

They decided to destroy this masterpiece of my work, appropriating the pieces while forgetting the unity which gives them their meaning and beauty, as well as their creative virtue. It is also not a coincidence if, equipped with a motley of tools whilst denying the spirit and vision that brought them into being, they were unable to discern the innovative work which was being reborn in front of their indifferent and disdainful gaze.

Finally, it is not by chance that following Deligne's initial impetus (which led him over the span of a few years toward the start of a new version of Hodge theory and the proof of the Weil conjectures), and despite his astounding abilities as well as my cohomology students' great means, I nowadays witness a "morose stagnation" in their domain, despite its prodigious richness and the multitude of things yet to be done.