《今天的播种 为了明天的收获》
国旗下讲话
敬爱的老师，亲爱的同学们：
早上好！今天国旗下讲话的题目是——今天的播种 为了明天的收获。 浓浓春意驱散了冬季的寒冷，万象更新孕育着春天的期待。今天我们又站在新的起跑线上，开始了新的人生征程。都说“一年之计在于春”，在这充满生机、充满希望的春天到来之际，同学们的心头也许正萌动着一种渴望与冲动，准备总结以往成功的经验与失败的教训，暗下决心，在新学期中珍惜时间，努力拼搏，以证明自己，超越自己，那么就让我们一起来为了美好的明天播种吧。 新学期，我们首先要播种“责任”，让生命多一份担当。因为责任，我们才会勇往直前；因为责任，我们才不轻言放弃；因为责任，我们才能最大可能地接近理想。学习之美，在于责任，同学友情刻骨铭心，父母之爱山高水长，老师教诲日月同辉，为了回报他人的关爱，为了不辜负亲人的期望，我们需要不懈地努力，需要尽到自己的一份责任。勤奋学习，成绩优秀是我们的责任，遵守校规校纪，热爱班集体是我们的责任，孝敬父母、报效国家更是我们的责任。 新学期，我们还要播种“执着”，让生命多一份坚强。“宝剑锋从磨砺出，梅花香自苦寒来”，学习之路充满坎坷，这就需要我们有明确的目标和科学的方法，需要我们有脚踏实地、不浮躁、不气馁的求学精神，“埋下头来学习，沉下心来读书”。学习之美，在于执着，篮球巨星迈克尔乔丹有一句名言：“我可以接受失败，但无法接受放弃。”在形形色色的诱惑面前，我们必须有“立根原在破崖中，咬定青山不放松”的坚毅，全神贯注，锲而不舍，才能笑在最后，笑得最好。
新学期，我们更要播种“理想”，让生命开出绚丽的花朵。“理想是石，敲出星星之火；理想是火，点燃熄灭的灯；理想是灯，照亮夜航的的路；理想是路，引你走到黎明。”学习之美，更在于理想，为了理想而奋斗、而拼搏，乐在其中，幸福亦在其中。
“责任”引领人生，“奋斗”实现价值。播种信念，收获行动；播种行动，收获习惯；播种习惯，收获性格；播种性格，收获命运。春天是耕耘的季节，只有辛勤地耕耘才能收获累累硕果。同学们，让我们用爱与智慧，踏踏实实地做好今天的每一件事，把平凡的事情做成经典，把简单的事情做得精彩，就一定能够迎来美好的明天。
我的讲话完毕，谢谢大家！
第二篇：收获与播种
Promenade Through a Life's Work: The Child and its Mother
1. The Magic of Things
When I was a child I loved going to school. The same instructor taught us reading, writing and arithmetic, singing (he played upon a little violin to accompany us), the archaeology of prehistoric man and the discovery of fire. I don't recall anyone ever being bored at school. There was the magic of numbers and the magic of words, signs and sounds, and the magic of rhyme, in songs or little poems. In rhyming there appeared to be a mystery that went beyond the words. I believed this until the day on which it was explained to me that this was just a 'trick': all one had to do in making a rhyme was to end two consecutive statements with the same syllable. As it by miracle, this turned ordinary speech into verse. What a revelation! In conversations at home I amused myself for weeks and months in spontaneously making verses. For awhile everything I said was in rhyme. Happily that's past. Yet even today, every now and then, I find myself making poems - but without bothering to search for rhymes when they do not arise spontaneously.
On another occasion a buddy who was a bit older than me, who was already going to the primary school, instructed me in negative numbers. This was another amusing game, yet one which lost its interest more quickly. And then there were crossword puzzles. I passed many a day in making them up, making them more and more complicated. This particular game combined the magic of forms with those of signs and words. Yet this new passion also passed away without a trace.
I was a good student in primary school, in Germany for the first year and then in France, although I wasn't what would be considered 'brilliant'. I became thoroughly absorbed in whatever interested me, to the detriment of all else, without concerning myself with winning the appreciation of the teacher. For my first year of schooling in France, 1940, I was interned with my mother in a concentration camp, at Rieucros, near Mende. It was wartime and we were foreigners - "undesirables" as they put it. But the camp administration looked the other way when it came to the children in the camp, undesirable or not. We came and left more or less as we wished. I was the oldest and the only one enrolled in school. It was 4 or 5 kilometers away, and I went in rain, wind and snow, in shoes if I was lucky to find them, that filled up with water.
I can still recall the first "mathematics essay", and that the teacher gave it a bad mark. It was to be a proof of "three cases in which triangles were congruent ". My proof wasn't the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of "gliding this figure over that one". It was self-evident that this man was unable or unwilling to think for himself in
judging the worth of a train of reasoning. He needed to lean on some authority, that of a book which he held in his hand. It must have made quite an impression on me that I can now recall it so clearly. Since that time, up to this very day, I've come to see that personalities like his are not the exception but the rule. I have lots to say about that subject in Récoltes et Semailles. Yet even today I continue to be stunned whenever I confront this phenomenon, as if it were for the first time.
During the final years of the war, during which my mother remained interned, I was placed in an orphanage run by the "Secours Suisse", at Chambon sur Lignon. Most of us were Jews, and when we were warned (by the local police) that the Gestapo was doing a round-up, we all went into the woods to hide for one or two nights, in little groups of two or three without concerning ourselves overmuch if it was good for our health. This region of the Cévennes abounded with Jews in hiding. That so many survived is due to the solidarity of the local population.
What struck me above all at the "Collège Cévenol" (where I was enrolled) was the extent to which my fellows had no interest in anything they were learning. As for myself I devoured all of my textbooks right from the beginning of each school year, convinced that this year, at last, we were really going to learn really interesting. Then for the rest of the year I had to figure out ways to employ my time as the program unfolded itself with tedious slowness over the course of the semester. However I should say that there were some really great teachers. Monsieur Friedal our instructor for Biology, was a man of high personal and intellectual qualities. However he was totally incapable of administering discipline, so that his class was in an interminable turmoil. So loud was the ruckus that it was impossible hear his voice rising above the din. No doubt that explains why I didn't become a biologist!
Much of my time, even during my lessons, (shh!..) was spent working on math problems. It wasn't long before the ones I found in the textbook were inadequate for me. This may have been because they all tended to resemble each other; but mostly because I had the impression that they were plucked out of the blue, without any idea of the context in which they'd emerged. They were 'book problems', not 'my problems'. However, there were questions that arose naturally. For example, when the lengths a, b, c of the three sides of a triangle are known, then the triangle itself is determined (up to its position in space), therefore there ought to be some explicit formula for expressing the area of that triangle as a function of a, b and c. The same had to be true for a tetrahedron when the 6 sides are known: what is its volume? That caused me no little difficulty, but in the end I did derive the formula after a lot of hard work. At any rate, once a problem "grabbed me", I stopped paying attention to the amount of time I had to spend on it, nor of all the other things that were being sacrificed for its sake does (This remain true to this day).
What I found most unsatisfactory in my mathematics textbooks was the absence of any serious attempt to tackle the meaning of the idea of the arc-length of a curve, or the
area of a surface or the volume of a solid. I resolved therefore to make up for this defect once I found time to do so. In fact I devoted most of my energy to this when I became a student at the University of Montpellier, between 1945 and 1948. The courses offered by the faculty didn't please me in the least. Although I was never told as much, I'd the impression that the professors had gotten into the habit of dictating from their texts, just like they used to do in the lycée at Mende. Consequently I stopped showing up at the mathematics department, and only did so to keep in touch with the official 'program'. For this purpose the textbooks were sufficient, but they had little to do with the questions I was posing, To speak truthfully, what they lacked was insight, even as the textbooks in the lycée were lacking in insight. Once delivered of their formulae for calculating lengths, areas, volumes in terms of simple, double or triple integrals (higher dimensions carefully avoided), they didn't care to probe further into the intrinsic meaning of these things. And this was as true of my professors as it was of the books from which they taught.
On the basis of my very limited experience I'd the impression that I was the only person in the entire world who was curious to know the answers to such mathematical questions. That was, at least, my private and unspoken opinion during all those years passed in almost total intellectual isolation, which, I should say, did not oppress me overmuch.(*) I don't think I ever gave any deep thought to trying to find out whether or not I was the only person on earth who considered such things important. My energies were sufficiently absorbed in keeping the promise I'd made with myself: to develop a theory that could satisfy me.
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(*)Between 1945 and 1948 my mother and I lived in a small hamlet about a dozen kilometers from Montpellier, named Mairargues (near Vendargues), surrounded by vineyards (My father disappeared in Auschwitz in 1942). We lived marginally on the tiny government stipend guaranteed to college students in France. Each year I participated in the grape harvests ("vendanges". Translators Note: I worked in these briefly, in the summer of 1970, in the region around Dijon.). After the harvests there was the gathering up of the loose remains of the grapes in the fields (grapillage), from which we made a more or less acceptable wine (apparently illegally). There was in addition, our garden, which, without having to do much work in it, furnished us with figs, spinach and even (in the late Fall ) tomatoes, which had been planted by a well-disposed neighbor right in the middle of a splendid field of poppies. It was 'the good life'. Although a little on the short side when it came to getting a new pair of glasses, or having to wear out one's shoes down to the soles. Luckily my mother, chronically invalided from her long term in the internment camps, had the right to free medical care. There was no way we could have paid for doctors.
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I never once doubted that I would eventually succeed in getting to the bottom of things, provided only that I took the effort to thoroughly review the things that came to me about them, and which I took pains to write down in black and white. We have, for
example, an undeniable intuition of volume. It had to be the reflection of some deeper reality, which for the moment remained elusive, but was ultimately apprehensible. It was this reality, plain and simple, that had to be grasped - a bit, perhaps, the way that the "magic of rhyme" had been grasped one day in a moment of understanding. In applying myself to this problem at the age of 17 and fresh out of the lycée, I believed that I could succeed in my objective in a matter of weeks. As it was, it preoccupied me fully three years. It even led me to flunk an examination, during my second year in college - in spherical trigonometry! (for an optional course on 'advanced astronomy') because of a stupid mistake in arithmetic. (I should confess here that I've always been weak in arithmetic, ever since leaving the lycée.)
Because of this I was forced to remain for a third year at Montpellier to obtain my license(*)
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(*)Translator's Note: the basic undergraduate degree in the French university system, not quite the same as our B.A.)
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, rather than heading immediately up to Paris - the only place, I was told, where one found people who really knew what was important in modern mathematics. The person who said this to me, Monsieur Soula, also assured me that all outstanding issues in mathematics had been stated and resolved, twenty or thirty years before, by a certain "Lebesgue"!(Translator's Italics). In fact he'd developed a theory of integration and measure (decidedly a coincidence!), beyond which nothing more needed to be said. Soula, it should be said, was my teacher for differential calculus, a good-hearted man and well disposed towards me. But he did not succeed at all in persuading me to his point of view. I must already have possessed the conviction that Mathematics has no limit in grandeur or depth. Does the sea have a "final end" ? The fact remains that at no point did it occur to me to dig out the book by Lebesgue that M.Soula had recommended to me, which furthermore he himself had never looked at! To my point of view, I could see little connection between what one might find in a book and the work I was doing to convince my own curiosity on issues that perplexed and intrigued me.
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2. The importance of Solitude
A few years after I finally established contact with the world of mathematics at Paris, I learned, among other things, that the work I'd in my little niche with the means at my disposal had (essentially) been long known to the whole world under the name of "Lebesgue's theory of measure and integration". In the eyes of my mentors, to whom
I'd described this work, and even shown them the manuscript, I'd simply "wasted my time", merely doing over again something that was "already known". But I don't recall feeling any sense of disappointment. At that time the very notion of "taking credit" for my own work, either to receive compliments or even the mere interest of anyone else, ) was furthest from my thoughts. My energies at that time were completely taken up with adjusting to a totally unfamiliar environment, above all with learning what one had to know to be treated like a mathematician.(*)
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(*)I talk briefly about this transitional period, which was rather rough, in the first part of Récoltes et Semailles(R&S I), in the section entitled "Welcoming the Stranger" (#9) --------------------------------------------------------------------------------
However, re-thinking those three years (1945-48), I realize that they weren't wasted in the least. Without recognizing it, I'd thereby familiarized myself with the conditions of solitude that are essential for the profession of mathematician, something that no-one can teach you. Without having to be told, without having to meet others who shared my thirst for understanding, I already knew "in my guts", that I was indeed a mathematician: because I knew that I was one who "makes mathematics", in the way someone "makes love". Quite simply, mathematics had become a mistress, ever receptive to gratifying my desire. These years of isolation laid the foundation for a faith that has never been shaken - neither by the discovery (arriving in Paris at the age of 20), of the full extent of my ignorance and the immensity of what I would be obliged to learn; nor (20 years later) by the turbulent events surrounding my final departure from the world of mathematics; nor, in recent years, by the thoroughly weird episodes of a metaphorical "Burial" of my person and my work, so perfectly orchestrated by those who were formerly my closest friends....
To state it in slightly different terms: in those critical years I learned how to be alone (*)
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(*) This formulation doesn't really capture my meaning. I didn't, in any literal sense learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these 3 years of work in isolation, when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring, in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law. I come back to this subject again in the note: "Roots and Solitude" (R&S IV, #171.3, in particular page 1080).
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By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member, or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me, both at the lyé and at the university, that one shouldn't bother worrying about what was really meant when using a term like "volume", which was "obviously self-evident", "generally
known", "unproblematic", etc. I'd gone over their heads, almost as a matter of course, even as Lesbesgue himself had, several decades before, gone over their heads. It is in this gesture of "going beyond", to be something in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one - it is in this solitary act that one finds true creativity. All others things follow as a matter of course. Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle - while for myself I felt clumsy. even oafish, wandering painfully up a arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone.
The infant has no trouble whatsoever being alone. It is solitary by nature, even when it's enjoying the company surrounding him or seeks his mother's tit when it is in need of it. And he is well aware, without having to be told, that the tit is for him, and knows how to use it. Yet all too often we have lost touch with the child within us. And it's often the case that we pass by the most important things without bothering to look at them...
If, in Récoltes et Semailles I'm addressing anyone besides myself, it isn't what's called a "public". Rather I'm addressing that someone who is prepared to read me as a person, and as a solitary person. It's to that being inside of you who knows how to be alone; it is to this infant that I wish to speak, and no-one else. I'm well aware that this infant has been considerably estranged.
It's been through some hard times, and more than once over a long period. It's been dropped off Lord knows where, and it can be very difficult to reach. One swears that it died ages ago, or that it never existed - and yet I am certain it's always there, and very much alive.
And, as well, I know how to recognize the signs that tell me I'm being understood. It's when, beyond all differences of culture and fate, what I have to say about my person finds an echo and an resonance in you, in that moment when you see, your own life, your own experience, in a light which, up to that moment, you'd not thought of paying attention to. It's not a matter of some sort of "re-identifying” something or someone that was lost to you. It means that you have rediscovered your own life, that which is closest to you, by virtue of the rediscovery that I've made of mine in the course of my writing these pages of R8#233coltes et Semailles, and even in those pages that I am in the process of setting down at this very moment.
3. The Interior Adventure - or Myth and Witnessing
Above all else, Récoltes et Semailles is a reflection on myself and only my life. At the same time, it is also a testimonial, and this in two ways. The testimonial on my past takes up the major portion of this reflection. Yet at the same time it is a testimonial to my immediate present - that is to say, up to the moment at which I'm writing it, in which the pages of Récoltes et Semailles are taking shape by the hour, night and day. These pages are the faithful witnesses of this long meditation on my life, as it is unfolding in real time, (and as it is unfolding even at this actual moment....)
These pages make no claim to literary excellence. They should be seen as a form of documentation on myself. I have refrained from touching it up in any way, (certainly not for stylistic reasons), save in a very restricted sense (*)
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(*) Thus, the rectification of mistakes (factual or interpretive) is not revised in the draft itself but appear as footnotes at the bottom of the page, or on those occasions when I return to the discussion of an earlier subject matter.
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If there is any affection on my part, it is the affectation of speaking the truth. And that's already quite a lot Furthermore one shouldn't look upon this document as some kind of "autobiography". You won't learn anything about my date of birth (which can only be of interest to someone engaged in casting my horoscope), nor the names of my father nor mother, or what they did in their lives, nor the name of my wife, or of other women who've been important in my life, or that of the children born from these loves, or what any of these people have done with their lives. It's not that these things haven't had their importance in my life, or have lost any of their importance. It is only that from the moment I began to work on this reflection I've felt under no compulsion to talk about these things directly, simply touching on them from time to time when they became relevant, nor have I felt impelled to cite names or vital statistics. It has never been my impression that doing so would add something meaningful to whatever I was engaged in examining at one time or another. (Thus, in the small selection of pages preceding this one I've included more of such details than in the 1000 pages
that follow it.
And, if you want to know, what is the "proposal" that I've laid out in over a thousand pages, my reply is: to tell the story, and by doing so to make the discovery of the interior adventure which has been and which continues to be the story of my life. This documentation-testament of my adventure is being conducted simultaneously on the two levels that I've speak about. There is first of all an exploration of the past adventure, its roots and origins in my childhood. And, secondly, there is the continuation and the rejuvenation of that "same" adventure, in line with the days and even the instants of the composition of Récoltes et Semailles, as a spontaneous response to a violent provocation into my life coming from the external world.(**) External events enrich this reflection only to the extent that they arouse a return to the interior adventure, or contribute to its clarification. Such a provocation has arisen from the long standing burial and plundering of my mathematical opus. It has aroused in my very powerful reactions of an unabashedly egocentric character, while at the same time revealed to me the profound ties which, unbeknownst to me continue to bind me to my opus
The fact that I happen to be one of the strong figures in modern mathematics does not, it is true, supply any reason why others should find my interior adventure interesting; nor does the fact that I'm on the outs with my colleagues after having totally changed my social environment and life style. Besides, there are any number of these colleagues, and even supposed friends, who don't hesitate, in public, to ridicule my so-called 'spiritual states'. What counts to them is 'results' and nothing else. The "soul", (which is to say that entity within us which experiences the "production" of these "results", or its direct effects, (such as the life of the "producer", as well as that of his associates)) is systematically despised, often with overtly promulgated derision.
Such attitudes are often labeled "humility"! To me this is merely a symptom of denial, of a strange sort of alienation, present in the very air we all breathe. It is a certainty that I don't write for the kind of person afflicted with this sort of disdain, who presumes to denigrate that which is the very best of what I have to offer him. A disdain, moreover, for what in fact determines his own life, as it has determined mine: those movements, superficial or profound, gross or subtle that animate the psyche, that very "soul" which lives experience and reacts upon it, which congeals or evaporates, which withdraws into itself or opens up...
The recital of an interior adventure can only be made by he who has lived it, and by none other. But, even if this recital has only been intended for one's own benefit, it is rare that it doesn't fall into the category of myth whose hero is the narrator. Such myths are born, not from the creative imagination of a culture or a people, but merely from the vanity of somehow who dare not accept a humbling reality, who has substituted for this reality some self-conceived fabrication. However, a true account, (if it is so) of an interior adventure as it has been truly lived is a precious thing. Not because of the prestige (rightly or wrong) that surrounds the narrator, but solely from
the fact that something with that degree of truthfulness really exists. Such a testament is priceless, whether it comes from a person deemed illustrious or notorious, or from some insignificant wage earner responsible for his family with little hope for the future, or even from a common criminal.
If this recitation of the facts has any value for others, it is to make them come face-to-face with their own selves, by means of an unvarnished testament of someone else's experience. Or, to state the case differently, to efface in himself, (even in the short time that it takes to read it), the contempt he holds for his own adventure, and for that "soul" which is both the passenger and the pilot.
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4. The novel of manners
In speaking of my mathematical past, and in the course of doing so uncovering (as if it were a matter of rescuing my own body) the mysterious turns taken by the colossal Burial of my life's work I have been led, without having intended it, to draw up a portrait of a certain milieu in a certain time in history - a time marked by the disintegration of certain timeless values which give meaning to all human endeavor. This is the aspect of the 'novel of manners', developing around a historical event which in no doubt unique in the "Annals of "Science? What has already been stated must make it clear that one shouldn't expect to find in Récoltes et Semailles, the "police report" or "dossier" of some celebrated "affair", written solely for the purpose of bringing one up to date. Any friend looking for such a report will go through it with his eyes closed, having seen nothing of any of the flesh and blood substance of Récoltes et Semailles.
As I explain, in much greater detail, in The Letter, the "police investigation"(or the "novel of manners") is to be found principally in Parts II and IV: "The Burial (1)-or the Robe of the Emperor of China? and: "The Burial (3) –or the Four Operations". In the course of writing these pages I have stubbornly brought to light a multitude of "juicy" findings, (to say the least), which I've attempted, for better or for worse, to "spruce up". Bit by bit I've found a coherent picture slowly emerging from the mists, one whose colors grow in intensity, one whose contours are becoming progressively sharper. In the notes that I've made on a daily basis, the "raw facts" which surface are inextricably mixed with personal reminiscences, comments and reflections on psychology, philosophy and even mathematics. That's the way it is and I can't do anything about it!
On the basis of the work already done, which has absorbed me for over a year, anyone wishing to extract a "dossier", in the mode of an investigative "wrap-up", will have to
spend many additional hours, if not days, depending on the interest or curiosity of the reader, in working it out. At one point I myself tried to extract such a dossier. This when I began the long footnote now known a
s "The Four Operations"(*)
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(*) What was intended as a footnote exploded into all of Part IV (with the same title of "The Four Operations"), comprising 70 notes stretching over 400 pages.
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Ultimately it wasn't possible. I failed totally! It's not my style, certainly not in my elderly years. In my present estimation, I've done enough, with the production of Récoltes et Semailles, for the benefit of the mathematics community to be able, without regrets, to leave for others (who may perhaps be found among my colleagues) the work of putting together the dossier it contains.
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5. The Inheritors and the Builder
The time has come to say a few words about my work in mathematics, something which at one point I held (and to my surprise still is) to be of some importance. I return more than once in Récoltes et Semailles to consider that work, sometimes in a manner that ought to be clear to everyone, though at other times in highly technical terms.(*)
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(*)Once in awhile one will discover, in addition to my observations about my past work, a discussion of some contemporary mathematical developments. The longest among these is in "The 5 photographs (Crystals and D-Modules)" in R&S IV, note #171 (ix)
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The latter passages will no doubt, for the most part, be 'over the heads' not only of the lay public, but also of those mathematical colleagues who aren't involved in this particular branch of mathematics. You are certainly more than welcome to skip any passages which impress you as being too 'specialized'. Yet even the layman may want to browse them, and by doing so perhaps be taken by the sense of a 'mysterious beauty' (as one of my non-mathematician friends has written) moving about within them like so many "strange inaccessible islands" in the vast and churning occasions of thought.
As I've often said, most mathematicians take refuge within a specific conceptual
framework, in a "Universe" which seemingly has been fixed for all time - basically the one they encountered "ready-made" at the time when they did their studies. They may be compared to the heirs of a beautiful and capacious mansion in which all the installations and interior decorating have already been done, with its living-rooms, its kitchens, its studios, its cookery and cutlery, with everything in short, one needs to make or cook whatever one wishes. How this mansion has been constructed, laboriously over generations, and how and why this or that tool has been invented (as opposed to others which were not),why the rooms are disposed in just this fashion and not another - these are the kinds of questions which the heirs don't dream of asking. It's their "Universe", it's been given once and for all! It impresses one by virtue of its greatness, (even though one rarely makes the tour of all the rooms) yet at the same time by its familiarity, and, above all, with its immutability.
When they concern themselves with it at all, it is only to maintain or perhaps embellish their inheritance: strengthen the rickety legs of a piece of furniture, fix up the appearance of a facade, replace the parts of some instrument, even, for the more enterprising, construct, in one of its workshops, a brand new piece of furniture. Putting their heart into it, they may fabricate a beautiful object, which will serve to embellish the house still further.
Much more infrequently, one of them will dream of effecting some modification of some of the tools themselves, even, according to the demand, to the extent of making a new one. Once this is done, it is not unusual for them make all sorts of apologies, like a pious genuflection to traditional family values, which they appear to have affronted by some far-fetched innovation.
The windows and blinds are all closed in most of the rooms of this mansion, no doubt from fear of being engulfed by winds blowing from no-one knows where. And, when the beautiful new furnishings, one after another with no regard for their provenance, begin to encumber and crowd out the space of their rooms even to the extent of pouring into the corridors, not one of these heirs wish to consider the possibility that their cozy, comforting universe may be cracking at the seams. Rather than facing the matter squarely, each in his own way tries to find some way of accommodating himself, one squeezing himself in between a Louis XV chest of drawers and a rattan rocking chair, another between a moldy grotesque statue and an Egyptian sarcophagus, yet another who, driven to desperation climbs, as best he can, a huge heterogeneous collapsing pile of chairs and benches!
The little picture I've just sketched is not restricted to the world of the mathematicians. It can serve to illustrate certain inveterate and timeless situations to be found in every milieu and every sphere of human activity, and (as far as I know) in every society and every period of human history. I made reference to it before, and I am the last to exempt myself: quite to the contrary, as this testament well demonstrates. However I maintain that, in the relatively restricted domain of
intellectual creativity, I've not been affected (*) by this conditioning process, which could be considered a kind of 'cultural blindness'? an incapacity to see (or move outside) the "Universe" determined by the surrounding culture.
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(*) The reasons for this are no doubt to be found in the propitious intellectual climate of my infancy up to the age of 5. With respect to this subject look at the note entitled "Innocence", (R&S III,# 107).
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I consider myself to be in the distinguished line of mathematicians whose spontaneous and joyful vocation it has been to be ceaseless building new mansions . (**)
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(**)This archetypal image of the "house" under construction appears and is elaborated for the first time in the note "Yin the Servant, and the New Masters"(R&S III #135)
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We are the sort who, along the way, can't be prevented from fashioning, as needed, all the tools, cutlery, furnishings and instruments used in building thenew mansion, right from the foundations up to the rooftops, leaving enough room for installing future kitchens and future workshops, and whatever is needed to make it habitable and comfortable. However once everything has been set inplace, down to the gutters and the footstools, we aren’t the kind of worker who will hang around, although every stone and every rafter carries the stamp of the hand that conceived it and put it in its place.
The rightful place of such a worker is not in a ready-made universe, however accommodating it may be, whether one that he's built with his own hands, or by those of his predecessors. New tasks forever call him to new scaffoldings, driven as he is by a need that he is perhaps alone to fully respond to. He belongs out in the open. He is the companion of the winds and isn't afraid of being entirely alone in his task, for months or even years or, if it should be necessary, his whole life, if no-one arrives to relieve him of his burden. He, like the rest of the world, hasn't more than two hands - yet two hands which, at every moment, know what they're doing, which do not shrink from the most arduous tasks, nor despise the most delicate, and are never resistant to learning to perform the innumerable list of things they may be called upon to do. Two hands, it isn't much, considering how the world is infinite. Yet, all the same, two hands, they are a lot....
I'm not up on my history, but when I look for mathematicians who fall into the lineage I'm describing, I think first of all of Evariste Galois and Bernhard Riemann in the previous century, and Hilbert at the beginning of this one. Looking for a representative among my mentors who first welcomed me into the world of mathematics (*), Jean Leray's name appears before all the others, even though my contacts with him have been very infrequent. (**)
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(*)I talk about these beginnings in the section entitled "The welcome stranger
"(ReS I, #9) (**) Even so I've been (following H. Cartan and J.P.Serre), one of the principal exploiters and promoters of one of the major ideas introduced by Leray, that of the bundle. It has been an indispensable tool in all of my work in geometry. It also provided me with the key for enlarging the conception of a (topological) space to that of a topos, about which I will speak further on.
Leray doesn't quite fill this notion that I have of a 'builder', in the sense of someone who 'constructs houses from the foundations up to the rooves.’ However, he's laid the ground for immense foundations where no one else had dreamed of looking, leaving to others the job of completing them and building above them or, once the house has been constructed, to set themselves up within its rooms (if only for a short time).....
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I've used large brush strokes in the making of my two sketches: that of the 'homebody" mathematician who is quite happy in adding a few ornaments to an established tradition, and that of the pioneer-builder (*), who cannot be restrained from crossing the 'imperious and invisible boundaries' that delimit a Universe (**)
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(*)Convenience has led me to form this hyphenated compound with a masculine resonance, "pioneer-builder' ("batisseur", and "pionnier"). These words express different phases in the impulse towards discovery whose connections are in fact too delicate to be satisfactorily expressed by them. A more satisfactory discussion will appear following this 'walking meditation', in the section "In search of the mother-or the two aspects' (#17)
(**)Furthermore, at the same time, and without intending to, he assigns to the earlier Universe (if not for himself then at least for his less mobile colleagues), a new set of boundaries, much enlarged yet also seemingly imperious and invisible than the ones he's replaced
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One might also call them using names that are perhaps less appropriate yet more suggestive, the "conservators" and the "innovators". Both have their motivations and their roles to play in the same collective adventure that mankind has been pursuing over the course of generations, centuries and millennia. In periods when an art or a science is in full expansion, there is never any rivalry between these two opposing temperaments (***). They differ yet are mutually complementary, like dough and yeast.
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(***)Such was the situation in mathematics during the period 1948-69 which I personally witnessed, when I was myself a part of that world. A period of reaction seems to have set in after my departure in 1970, one might call it a "consensual scorn" for 'ideas' of any sort, notably for those which I had introduced.
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Between these two types at the extremes (though there is no opposition in nature between them) one finds a spectrum of every kind of person. A certain "homebody" who cannot imagine that he will ever leave his familiar home territory, or even contemplate the work involved in setting up somewhere else, will all the same put his hand to the trowel for digging out a cellar or an attic, add on another story, even go so far as to throw up the walls for a new, more modest, building next to his present one.(****)
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(****) Most of my mentors (to whom I devote all my attention in "A welcome debt", Introduction, 10), have this in-between temperament. I'm thinking in particular of Henri Cartan, Claude Chevalley, André Weil, Jean-Pierre Serre, and Laurent Schwartz. With the exception perhaps of Weil, they all, at last, cast an "auspicious eye" without "anxiety or private disapproval" at the lonely adventures in which I was engaged. --------------------------------------------------------------------------------
Without having the character of a true builder, he will frequently express sympathy for one who does, or at least feel no anxiety or private disapproval towards one who has shared the same dwelling with him, even when he does strange things like setting up pillars and building blocks in some outlandish setting, with the attitude of someone who already sees a palace in front of him.
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6. Visions and Viewpoints
But I must return to myself and my work.
If I have excelled in the art of the mathematician, it is due less to my facility or my persistence in working to find solutions for problems delegated to me by my predecessors, than to the natural propensity which drives me to envisage questions, ones that are clearly critical, which others don't seem to notice, and to come up with "good ideas" for dealing with them (while at the same time no-one else seems to suspect that a new idea has arrived), and "original formulations" which no-one else has imagined. Very often, ideas and formulations interact in so effective a manner that the thought that they might be incorrect does not arise, (apart from touching them up a bit). Also as well, when it’s not a matter of putting the pieces together for publication, I take the time to go further, or to complete a proof which, once the formulation and its
context have been clarified, is nothing more than what is expected of a true "practitioner", if not simply a matter of routine. Numberless things command our attention, and one simply cannot follow all of them to the end! Despite this it is still the case that the theorems and propositions in my written and published work that are cast into the proper form of a demonstration number in the thousands. With a tiny number of exceptions they have all joined the patrimony of things accepted as "known" by the community, and are used everywhere.
Yet, even more than in the discovery of new questions, notions and formulations, my unique talent appears to consist of the entertainment of fertile points of view which lead me to introduce and to, more or less, develop completely original themes. It is that constitutes my most essential contribution to the mathematics of my time. To speak frankly, these innumerable questions, notions and formulations of which I've just spoken, only make sense to me from the vantage of a certain 'point of view' - to be more precise, they arise spontaneously through the force of a context in which they appear self-evident: in much the same way as a powerful light (though diffuse) which invades the blackness of night, seems to give birth to the contours, vague or definite, of the shapes that now surround us. Without this light uniting all in a coherent bundle, these 10 or 100 or 1000 questions, notions or formulations look like a heterogeneous yet amorphous heap of "mental gadgets", each isolated from the other- and not like parts of a totality of which, though much of it remains invisible, still shrouded in the folds of night, we now have a clear presentiment.
The fertile viewpoint is that which reveals to us, as so many parts of the same whole that surrounds them and gives meaning to them, those burning questions that few are aware of, (perhaps in response to these questions) thoroughly natural notions yet which none had previously conceived, and formulations which seem to flow from a common source, which none had dared to pose despite their having been suggested for some time by these questions, and, and for which the ideas had yet to emerge. Far more indeed, than what are called the "key theorems:" of mathematics, it is these fertile viewpoints which are, in our particular craft(*) the most powerful tools for discovery- rather they are not tools exactly, but the very eyes of the researcher who, in a deeply passionate sense, wishes to understand the nature of mathematical reality. --------------------------------------------------------------------------------
(*) This is not only the case in "our art", but, so it seems to me, in all forms of discovery, at least in the domain of intellectual knowledge/
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Thus, the fertile viewpoint is nothing less than the "eye" which, at one and the same time, enables us to discover and, at the same time, recognize the simple unity behind the multiplicity of the thing discovered. And, this unity is, veritably, the very breath of life that relates and animates all this multiplicity.
Yet, as the word itself suggests, a "viewpoint" implies particularity. It shows us but a single aspect of a landscape or a panorama out of a diversity of others which are equally valuable, and equally "real". It is to the degree that the complementary views
of the same reality cooperate, with the increasing population of such "eyes", that one's understanding of the true nature of things advances. The more complex and rich is that reality that we wish to understand, the more the necessity that there be many "eyes" (**) for receiving it in all its amplitude and subtlety.
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(*)Every viewpoint entails the development of a language appropriate to itself for its expression. To "have several eyes", or several "viewpoints" for comprehending a certain situation, also requires (at least in mathematics), that one has at one's disposal several distinct languages with which to grasp it.
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And it often happens that a light-beam composed of many viewpoints focusing on a single immense landscape, by virtue of that gift within us which can apperceive the One within the diversity of the Many, gives birth to something entirely new; to something which transcends each of the partial perspectives, in the same way that a living organism transcends its appendages and organs. This new thing may be named a Vision It is vision which unites the various viewpoints that compose it, while revealing to us other viewpoints which up to then had been ignored, even as the fertile viewpoint permits one to both discover and apprehend as part of a single Unity, a multiplicity of new questions, notions and formulations.
Otherwise stated: Vision is, to the viewpoints from which it springs, and which it unites, like the clear, warm light of day is to the different frequencies of the solar spectrum. A vision that is both extensive and profound is like an inexhaustible wellspring, made to inspire and illuminate the work, not only of the person in whom it first sees the light of day and becomes its servant, but that of generations, fascinated perhaps (as he was also) by those distant boundaries which it opens up.
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7. "The Great Idea - or the forest and the trees"
The so-called "productive period" of my mathematical activity, which is to say the part that can be described by virtue of its properly vetted publications, covers the period from 1950 to 1979, that is to say 20 years. And, over a period of 25 years, between 1945 (when I was 17), and 1969, (approaching my 42nd year), I devoted virtually all of my energy to research in mathematics. An exorbitant investment, I would agree. It was paid for through a long period of spiritual stagnation, by what one may call an burdensome oppression which I evoke more than once in the pages of Récoltes et
Semailles. However, staying strictly within the limited field of purely intellectual activity, by virtue of the blossoming forth and maturation of a vision restricted to the world of mathematics alone, these were years of intense creativity
During this lengthy period of my life, the greater part of my energy was consecrated to what one might call "piece work": the scrupulous work of shaping, assembling, getting things to work, all that was essential for the construction of all the rooms of the houses, which some interior voice (a demon perhaps?) exhorted me to build, the voice of a master craftsman whispering to me now and then depending on the way the work was advancing. Absorbed as I was by the tasks of my craft- brick-layer, stone-mason, carpenter, plumber, metal worker, wood worker - I rarely had the time to write down in black and white, save in sketching the barest outlines, the invisible master-plan that except, (as it became abundantly clear later) to myself underlined everything, and which, over the course of days, months and years guided my hand with the certainty of a somnambulist. (*)
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(*)The image of the "somnambulist" is inspired by the title of the remarkable book by Arthur Koestler, "The Sleepwalkers" (published in France by Calman Levy), subtitled, "A history of conceptions of the universe" from the origins of scientific thought up to Newton. An aspect of this history which particularly impressed Koestler was the extent to which, so often, the road leading from one point of our knowledge of the world to some other point, seemingly so close (and which appears in retrospect so logical), passes through the most bizarre detours almost to the point of appearing insane; and how, all the same, through these thousand-fold detours in which one appears to be forever lost, and with the certainty of "Sleepwalkers", those persons devoted to the search for the "keys" to the Universe fall upon, as if in spite of themselves and without always being aware of it, other "keys" which they did not anticipate, yet which prove in the long run to be the correct ones.
On the basis of what I've been able to see around me at the level of mathematical discovery, these incredible detours of the roads of discovery are characteristic of certain great investigators only. This may be due to the fact that over the last two or three centuries the natural sciences, and mathematics even more so, have gradually liberated themselves from all the religious and metaphysical assumptions of their culture and time, which served as particularly severe brakes on the universal development, (for better or worse) of a scientific understanding of the universe. It is true, all the same, that some of the most basic and fundamental notions in mathematics (such as spatial translation, the group, the number zero, the techniques of calculus, the designation of coordinates for a point in space, the notion of a set, of a topology, without even going into negative and complex numbers), required millennia for their emergence and acceptance. These may be considered so many eloquent signs of that inherent "block", implanted in the human psyche, against the conceptualization of totally new ideas, even when these ideas possess an almost infantile simplicity, and which one would think would be obvious based on the available evidence, over generations, not to say millennia....
To return to my own work, I've the impression that the "hand waving" (perhaps more numerous than those of my colleagues), has been largely over matters of detail, usually quickly rectified by my own careful attention. These might be called simple "accidents of the road" of a purely local character without any serious effects on the validity of the underlying intuitions of the specific situation. On the other hand, at the level of ideas and large-scale intuitions, I feel that my work stands the test of time, as incredible as that may seem. It is this certainty without hesitation of having grasped at every instant, if not exactly the ends to which my thought leads, (which often enough lie hidden), but at least the most fertile directions which ought to be explored that will lead directly to that which is most essential. It is this quality of "certitude" which has brought to my mind Koestler's image of the "sleepwalker"
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It must be said that all of this piecework to which I've devoted such loving attention, was never in the least disagreeable. Furthermore, the modes of mathematical expression promoted and practiced by my mentors gave pre-eminence (to say the least!) to the purely technical aspect of the work, looking askance at any "digressions" that would appear to distract one from his narrow "motivations", that is to say, those which might have risked bringing out of the fogs some inspiring image or vision but which, because it could not be embodied right away into tangible forms of wood, stone or cement, where treated more appropriate to the stuff of dreams rather than the work of the conscientious or dedicated artisan.
In terms of its quantity, my work during these productive years found its concrete expression in more than 12,000 published pages in the form of articles, monographs or seminars(*)
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(*)Starting with the 60's a portion of these publications were written in collaboration with colleagues (primarily J. Dieudonné) and students.
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, and by hundreds, if not thousands of original concepts which have become part of the common patrimony of mathematics, even to the very names which I gavethem when they were propounded.(**)
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(**)The most significant of these ideas have been outlined in the Thematic Outline (Esquisse Thématique), and in the Historical Commentary that accompanies it, included in Volume 4 of the Mathematical Reflections. Some of their labels had been suggested to me by students or friends, such as the term "smooth morphism"(morphisme lisse) (J. Dieudonné) or the combine "site, stack, sheaf, connection" (?"site, champ, gerbe, lien") developed in the thesis of Jean Giraud. --------------------------------------------------------------------------------
In the history of mathematics I believe myself to be the person who has introduced the greatest number of new ideas into our science, and at the same time, the one who has therefore been led to invent the greatest number of terms to express these ideas accurately, and in as suggestive a manner as possible.
These purely "quantitative" indicators give no more, admittedly, than a crude overview of my work, to the total neglect of those things which gave it life, soul and vigor. As I've written above, the best thing I've brought to mathematics has been in terms of original viewpoints which I've first intuited, then patiently unearthed and developed bit by bit. Like the notions I've mentioned, these original viewpoints, which introduced into a great multiplicity of distinct situations, are themselves almost without limit.
However, some viewpoints are more extensive than others, which along have the capacity to encapsulate a multitude of other partial viewpoints, in a multitude of different particular instances. Such viewpoints may be characterized as "Great Ideas". By virtue of their fecundity, an idea of this kind give birth to a teeming swarm of progeny, of ideas inheriting its fertility, which, for the most part,(if not all of them) do not have as extensive a scope as the mother-concept.
When it comes to presenting a "Great Ideas", to "speak it", one is faced with, almost always, a problem as delicate as its very conception and slow gestation in the person who has conceived it - or, to be more precise, that the sum total laborious work of gestation and formation is the "expression" of the idea: that work which consists of patiently bringing it to light, day after day, from the mists that surround its birth, to attain, little by little, some tangible form, in a picture that is progressively enriched, confirmed and refined over the course of weeks, months and years. Merely to name the idea in terms of some striking formulation, or by fairly technical key words, may end up being a matter of a few lines, or may extend to several pages. Yet it is very rare to find anyone who, without knowing it in advance, is able to "hear" this "name", or recognize its face. Then, when the idea has attained to its full maturity, one may be able to express it in a hundred or so pages to the full satisfaction of the worker in whom it had its birth. Yet it may also be the case that even a thousand pages, extensively reworked and thought over, will not suffice to capture it.(*)
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(*)When I left the world of mathematics in 1970, the totality of my publications (many of which were collaborations) on the central theme of schemas came to something like ten thousand pages. This, however, constitutes only a modest portion of a gigantic program that I envisaged about schemas. This program was abandoned sine die with my departure, and that despite the fact that, apart from minor and inconsequential matters, everything that had already been developed and published was available to everyone, and had entered into the common heritage of notions and results normally deemed to be "well known."
That piece of my program on the theme of schemas, their prolongations and their ramifications, that I'd completed at the time of my departure, represents all by itself the greatest work on the foundations of mathematics ever done in the whole history of mathematics (Italics added by the translator so that there should be no misunderstanding of who is speaking), and undoubtedly one of the greatest achievements in the whole history of Science.
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And, in one case as in the other, among those who, in order to make it their own, have become acquainted with the work involved in bringing the idea to its full presentation, like a great forest that has miraculously sprung up in a desert- I would dare to bet that there are many among them who will, seeing all these healthy and vigorous trees, be inspired to avail themselves of them (whether for climbing, to fabricate planks and pillars, or to feed the fires in their hearths....) Yet there are few indeed who ever get to see the forest...
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8. The Vision-or 12 Themes for a Harmonization
Perhaps one might say that a "Great Idea" is simply the kind of viewpoint which not only turns out to be original and productive, but one which introduces into a science an extraordinary and new theme. Every science, once it is treated not as an instrument for gaining dominion and power, but as part of the adventure of knowledge of our species through the ages, may be nothing but that harmony, more or less rich, more or less grand depending on the times, which unfolds over generations and centuries through the delicate counterpoint of each of its themes as they appear one by one, as if summoned forth from the void to join up and intermingle with each other.
Among the numerous original viewpoints which I've uncovered in Mathematics I find twelve which, upon reflection, I would call "Great Ideas". (*)
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(*) For the sake of the mathematical reader, here is the list of these 12 master ideas, or "master-themes" of my work, in chronological order:
Topological Tensor Products and Nuclear Spaces
"Continuous" and "Discrete" dualities (Derived Categories, the "6 operations")
The Riemann-Roch-Grothendieck Yoga (K-Theory and its relationship to Intersection Theory)
Schemes
Topos Theory
Etale Cohomology and l-adic Cohomology
Motives, Motivic Galois Groups (*-Grothendieck categories)
Crystals, Crystalline Cohomology, yoga of the DeRham coefficients, the Hodge c oefficients
"Topological Algebra":(infinity)-stacks;derivations;cohomological formalism of topoi, insipiring a new conception of homotopy.
Mediated topology
The yoga of un-Abelian Algebraic Geometry. Galois-Teichmüller Theory
Schematic or Arithmetic Viewpoints for regular polyhedra and in general all regular configurations. Apart from the themes in item 12, a goodly portion of which first appeared in my thesis of 1953 and was further developed in the period in which I worked in functional analysis between 1950 and 1955, the other eleven themes were discovered and developed during my geometric period, starting in 1955
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To appreciate my work as a mathematician, to "sense it", is to appreciate and to sense, as best one can a certain number of its ideas, together with the grand themes they introduce which form the framework and the soul of the work.
In the nature of things, some of these ideas are "grander than others"!, others "smaller". In other words, among these new and original themes, some have alarger scope, while others delve more deeply into the mysteries of mathematical verities. (**).
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(**)To give some examples, the idea of greatest scope appears to me to be that of the topos, because it suggests the possibility of a synthesis of algebraic geometry, topology and arithmetic. The most important by virtue of the reach of those developments which have followed from it is, at the present moment, the schema. (With respect to this subject see the footnotes from to the previous section (#7)) It is this theme which supplies the framework, par excellence, of 8 of the others in the above list. (that is to say, all the others except 1,5 and 10), which at the same time furnishing the central notion fundamental to a total reformation, from top to bottom, of algebraic geometry and of the language of that subject.
At the other extreme, the first and last of these 12 themes are of much less significance. However, vis-a-vis the last one, having introduced a new way of looking at the very ancient topic of the regular polyhedra and regular configurations in general, I am not sure that a mathematician who gives his whole life to studying them will have wasted his time. As for the first of these themes, topological tensor products, it has played the
role of a handy tool, rather than as the springboard for future developments deriving from it. Even so I've heard, particularly in recent years, sporadic echoes of research resolving (20 or 30 years later!) some of the issues that were left open by my discoveries.
Among the 12 themes, the deepest is that of the motifs, which are closely tied to those of an-Abelian Algebraic Geometry, and that of Galois-Teichmüller Yoga.
In terms of the effectiveness of the tools I've created, laboriously polished and brought to perfection, now heavily used in certain "specialized research areas" in the last 2 decades, I would single out schemas and étale and l-adiccohomologies. For the well-informed mathematician I would claim that, up to the present moment, it can scarcely be doubted that these schematic tools. suchas l-adic cohomology, etc., figure among the greatest achievements of this century, and will continue to nourish and revitalize our science in all following generations.
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Among these grand ideas one finds 3 (and hardly the least among them) which, having appeared only after my departure from the world of mathematics, are still in a fairly embryonic state: they don't even exist "officially", since they haven't appeared in any publication, (which one might consider the equivalent of a birth certificate)(*). --------------------------------------------------------------------------------
(*)The only "semi-official" text in which these three themes are sketched, more or less, is the Outline for a Program, edited in January 1984 on request from a unit of CNRS. This text (which is also discussed in section 3 of the Introduction, "Compass and Luggage"), should be, in principle, included in volume 4 of Mathematical Reflections. --------------------------------------------------------------------------------
The twelve principal themes of my opus aren't isolated from each other. To my eyes they form a unity, both in spirit and in their implications, in that one finds in them a single persistent tone, present in both "officially published" and "unpublished" writings. Indeed, even in the act of writing these lines I seem to recapture that same tone- like a call! - persisting through 3 years of "unrewarded" work, in dedicated isolation, at a time when it mattered little to me that there were other mathematicians in the world besides myself, so taken was I by the fascination of what I was doing...This unity does not derive alone as the trademark of a single worker. The themes are interconnected by innumerable ties, both subtle and obvious, as one sees in the interconnection of differing themes, each recognizable in its individuality, which unfold and develop in a grand musical counterpoint- in the harmony that assembles them together, carries them forward and assigns meaning to all of them, a movement and wholeness in which all are participants. Each of these partial themes seems to have been born out of an all-engulfing harmony and to be reborn from one instant to the next, while at the same time this harmony does not appear as a mere "sum" or "resultant" of all the themes that make it up, that in some sense are pre-existent within it. And, to speak truly, I cannot avoid the feeling (cranky as it must appear), that in some sense it is actually this "harmony", not yet present but which already "exists'
somewhere in the dark womb of things awaiting birth in their time - that it is this and this alone which has inspired, each in its turn, these themes which acquire meaning only through it. And it is that harmony which called out to me in a low and impatient voice, in those solitary and inspired years of my emergence from adolescence....
It remains true that these 12 master-themes of my work appear, as through a kind of secret predestination, to abide concurrently within the same symphony -or, to use a different image, each incarnates a different "perspective" on the same immense vision.
This vision did not begin to emerge from the shades, or take recognizable shape, until around the years 1957, 1958 - years of enormous personal growth. (*)
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(*)1957 was the year in which I began to develop the theme "Riemann-Roch" (Grothendieck version) - which almost overnight made me into a big "movie star". It was also the year of my mother's death and thereby the inception of a great break in my life story. They figure among the most intensely creative years of m
y entire life, not only in mathematics. I'd worked almost exclusively in mathematics for 12 years. In that year there was the sense that I'd perhaps done what there was to do in mathematics and that it was time to try something else. This came out of an interior need for revitalization, perhaps for the first time in my life. At that time I imagined that I might want to be a writer, and for a period of several months I stopped doing mathematics altogether. Finally however I decided to return, just long enough to give a definitive form to the mathematical works I'd already done, something I imagined would take only a few months, perhaps a year at most...
The time wasn't ripe, apparently, for a complete break. What is certain is that in taking up my work in mathematics again, it took possession of me, and didn't let go of me for another 12 years!
The following year (1958) is probably the most fertile of all my years as a mathematician. This was the year which saw the birth of the two central themes of the new geometry through the launching of the theory of schemes (the subject of my paper at the International Congress of Mathematicians at Edinborough in the summer of that year) and the appearance of the concept of a "site", a provisional technical form of the crucial notion of the topos. With a perspective of thirty years I can say now that this was the year in which the very conception of a new geometry was born in the wake of these two master-tools: schemes (metamorphosed from the anterior notion of the "algebraic variety") and the topos (a metamorphoses, even deeper, of the idea of space).
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It may appear strange, but this vision is so close to me and appeared so "self-evident", that it never occurred to me until about a year ago to give a name to it. (*)
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(*)It first occurred to me to name this vision in the meditation of December 4th, 1984, (in subnote #136-1) to the footnote "Yin the Servant"(2) - or Generosity" (Récoltes et
Semailles, pg. 637)
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(Although it is certainly one of my passions to be constantly giving names to things that I've discovered as the best way to keep them in mind...) It is true that I can't identify a particular moment at which this vision appeared, or which I can reconstruct through recollection. A new vision of things is something so immense that one probably can't pin it down to a specific moment, rather it takes possession of one over many years, if not over several generations of those persons who examine and contemplate it. It is as if new eyes have to be painfully fashioned from behind the eyes which, bit by bit, they are destined to replace. And this vision is also too immense for one to speak of "grasping" it, in the same way that one "grasps" an idea that happens to arise along the way. That's no doubt why one shouldn't be surprise that the idea of giving a name to something so enormous, so close yet so diffuse, only occurred to me in recollection, and then only after it had reached its full maturity. In point of fact, for the next two years my relationship to mathematics was restricted (apart from teaching it) to just getting it done- to giving scope to a powerful impulse that ceaselessly drew me forward, into an "unknown" that I found endlessly fascinating. The idea didn't occur to me to pause, even for the space of an instant, to turn back and get an overview of the path already followed, let alone place it in the context of an evolving work. (Either for the purpose of placing it in my life, as something that continued to attach me to profound and long neglected matters; or to situate it in that collective adventure known as "Mathematics")
What must appear even more strange, in order to get me to stop for a moment and re-establish acquaintance with these half-forgotten efforts, (or to think of giving a name to the vision which is its heart and soul), I had to face a confrontation with a "Burial" of gigantic proportions: with the burial, by silence and derision, of that vision and of the worker who conceived it...
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9. Structure and Form - or the Voice of Things"
Without intention on my part this "Avant-Propos" is turning, bit by bit, into a kind of formal presentation of my opus, designed above all for the non-mathematical reader. I'm too involved by now to change orientations, so I'll just plug ahead and try to bring all these "presentations" to an end! All the same, I'd like to say at least a few words on
the substance of these "fabulous great ideas" (otherwise called "master themes") which I've depicted in the preceding pages, as well as something about the nature of this proclaimed "vision" within which these master themes are floating about. Without availaling myself of a highly technical language the most I can do is invoke the image of an intense sort of flux (if in fact one can speak of 'invoking' something)....(*) --------------------------------------------------------------------------------
Although this image must remain "fluid" does not mean that it isn't accurate, or that it doesn't faithfully convey something essential of the thing contemplated (in this case my opus). Conversely, it is possible to make a representation of something that is static and clear that can be highly distorted or touch only on its superficial aspects. Therefore, if you are "taken" by what I see as the essence of my work, (and something of that "image" abiding in me must have been communicated to you), you can flatter yourself to have grasped more about it than any of my learned colleagues!
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It is traditional to distinguish three kinds of "qualities" or "aspects" of things in the Universe which adapt themselves to mathematical reflections. These are (1) Number (**); (2) Magnitude and (3) Form
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By this is meant the "natural numbers" o,1,2,3, etc., or (at most) the numbers (such as rational fractions) which are expressed in terms of them by the elementary operations. These numbers cannot, (as can the "real numbers") be used to measure quantities subject to continuous variation, such as the distance between two arbitrary points on a straight line, in a plane or in space.
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One can also speak of them as the "arithmetical aspect", the "metric aspect" and the "geometric aspect" of things. In most of the situations studied in mathematics, these three aspects are simultaneously present in close interaction. Most often, however, one finds that one or another of them will predominate. It's my impression that for most mathematicians its quite clear to them (for those at least who are in touch with their own work) if they are "arithmeticians", "analysts", or "geometers", and this remains the case no matter how many chords they have on their violin, or if they have played at every register and diapason imaginable. My first solitary reflections, on Measure Theory and Integration, placed me without ambiguity under the rubrique of Analysis. And this remained the same for the first of the new themes that I introduced into mathematics, (which now appears to me to be of smaller dimensions than the 11 that followed). I entered mathematics with an "analytic bias", not because of my natural temperament but owing to "fortuitous circumstances": it was because the biggest gap in my education, both at the lycée and at the university, was precisely in this areaof the "analytic aspect" of things.
The year 1955 marked a critical departure in my work in mathematics: that of my passage from "analysis" to "geometry". I well recall the power of my emotional response (very subjective naturally); it was as if I'd fled the harsh arid steppes to find myself suddenly transported to a kind of "promised land" of superabundant richness,
multiplying out to infinity wherever I placed my hand in it, either to search or to gather... This impression, of overwhelming riches has continued to be confirmed and grow in substance and depth down to the present day.(*)
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(*) The phrase "superabundant richness" has this nuance: it refers to the situation in which the impressions and sensations raised in us through encounter with something whose splendor, grandeur or beauty are out of the ordinary, are so great as to totally submerge us, to the point that the urge to express whatever we are feeling is obliterated.
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That is to say that, if there is one thing in Mathematics which (no doubt this has always been so) fascinates me more than anything else, it is neither "number", nor "magnitude" but above all "form". And. among the thousand and one faces that form chooses in presenting itself to our attention, the one that has fascinated me more than any other, and continues to fascinate me, is the structure buried within mathematical objects. One cannot invent the structure of an object. The most we can do is to patiently bring it to the light of day, with humility - in making it known it is "discovered". If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we aren't in any sense "making" or "building" these structures. They hardly waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we've been detecting or discovering, to deliver up that reticent structure, which we can only grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to invent the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to "construct" with the help of this language, bit by bit, those "theories" which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them, by a language in a constant state improvement, and constantly in a process of recreation, under the pressure of immediate necessity.
As the reader must have realized by now, these "theories", "constructed out of whole cloth", are nothing less than the "stately mansions" treated in previous sections: those which we inherit from our predecessors, and those which we are led to build with our own hands, in response to the way things develop. When I refer to "inventiveness" (or imagination) of the maker and the builder, I am obliged to adjoin to that what really constitutes it soul or secret nerve. It does not refer in any way to the arrogance of someone who says "This is the way I want things to be!" and ask that they attend him at his leisure, the kind of lousy architect who has all of his plans readymade in his head without having scouted the terrain, investigated the possibilities and all that is required.
The sole thing that constitutes the true "inventiveness" and imagination of the researcher is the quality of his attention as he listens to the voices of things. For
nothing in the Universe speaks on its own or reveals itself just because someone is listening to it. And the most beautiful mansion, the one that best reflects the love of the true workman, is not the one that is bigger or higher than all the others. The most beautiful mansion is that which is a faithful reflection of the structure and beauty concealed within things.
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10: The new Geometry: or, the Marriage of Number and Magnitude.
But here I am, digressing again! I set out to talk about the "master-themes", with the intention of unifying them under one "mother vision", like so many rivers returning to the Ocean whose children they are..This great unifying vision might be described as a new geometry. It appears to be similar to the one that Kronecker dreamed of a century ago (*)
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(*) I only know about "Kronecker's Dream" through hearsay, in fact it was when somebody (I believe it was John Tate) told me that I was about to carry it out. In the education which I received from my elders, the historical references were very rare indeed. I was nourished, not by reading the works of others, ancient or modern, but above all through communication, through conversations or exchanges of letters, with other mathematicians, beginning with my teachers. The principal, perhaps the only external inspiration for the sudden and vigorous emergence of the theory of schemes in 1958, was the article by Serre commonly known by its label FAC (Faisceaux algébriques cohérents) that came out a few years earlier. Apart from this, my primary source of inspiration in the development of the theory flowed entirely from itself, and restored itself from one year to the next by the requirements of simplicity and internal coherence, and from my effort at taking into account in this new context, of all that was "commonly known" in algebraic geometry (which I assimilated bit by bit as it was transformed under my hands), and from all that this "knowledge" suggested to me. But the reality is (which a bold dream may sometimes reveal, or encourage us to discover) surpasses in every respect in richness and resonance even the boldest and most profound dream. Of a certainty, for more than one of these revelations of the new geometry, (if not for all of them), nobody, the day before it appeared, could have imagined it - neither the worker nor anyone else.
One might say that "Number" is what is appropriate for grasping the structure of
"discontinuous" or "discrete" aggregates. These systems, often finite, are formed from "elements", or "objects" conceived of as isolated with respect to one another. "Magnitude" on the other hand is the quality, above all, susceptible to "continuous variation", and is most appropriate for grasping continuous structures and phenomena: motion, space, varieties in all their forms, force fields, etc. Thereby, Arithmetic appears to be (overall) the science of discrete structures while Analysis is the science of continuous structures.
As for Geometry, one can say that in the two thousand years in which it has existed as a science in the modern sense of the word, it has "straddled" these two kinds of structure, "discrete" and "continuous". (*)
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(*)In point of fact, it has traditionally been the "continuous" aspect of things which has been the central focus of Geometry, while those properties associated with "discreteness", notably computational and combinatorial properties have been passed over in silence or treated as an after-thought. It was therefore all the more astonishing to me when I made the discovery, about a dozen years ago, of the combinatorial theory of the Icosahedron, even though this theory is barely scratched (and probably not even understood) in the classic treatise of Felix Klein on the Icosahedron. I see in this another significant indicator of this indifference (of over 2000 years) of geometers vis-a-vis those discrete structures which present themselves naturally in Geometry: observe that the concept of the group (notably of symmetries) appeared only in the last century (introduced by Evariste Galois), in a context that was considered to have nothing to do with Geometry. Even in our own time it is true that there are lots of algebraists who still haven't understood that Galois Theory is primarily, in essence, a geometrical vision, which was able to renew our understanding of so-called "arithmetical" phenomenon.
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For some time in fact one can say that the two geometries considered to be distinct species, the discrete and the continuous, weren't really "divorced". They were rather two different ways of investigating the same class of geometric objects: one of them accentuated the "discrete" properties (notably computational and combinatorial) while the other concerned itself with the "continuous" properties (such as location in an ambient space, or the measurement of "magnitude" in terms of the distances between points, etc.)
It was at the end of the last century that a divorce became immanent, with the arrival and development of what came to be called" Abstract (Algebraic) Geometry". Roughly speaking, this consisted of introducing, for every prime number p, an algebraic geometry "of characteristic p", founded on the model (continuous) of the Geometry (algebraic) inherited from previous centuries, however in a context which appeared to be resolutely "discontinuous", or "discrete". This new class of geometric objects has taken on a growing significance since the beginning of the century, in particular owing to their close connections with arithmetic, which is the science par excellence of discrete structures. This appears to be one of the notions motivating the work of
André Weil (**), perhaps the driving force (which is usually implicit or tacit in his published work, as it ought to be): the notion that "the" Geometry (algebraic), and in particular the "discrete" geometries associated with various prime numbers, ought to supply the key for a grand revitalization of Arithmetic.
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(**)André Weil, a French mathematician who emigrated to the United States, is one of the founding members of the "Bourbaki Group", which is discussed in some length in the first part of Récoltes et Semailles (as is Weil himself from time to time).
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It was with this perspective in mind that he announced, in 1949, his famous "Weil conjectures". These utterly astounding conjectures allowed one to envisage, for these new " discrete varieties" (or "spaces"), the possibility for certain kinds of constructions and arguments (*) which up to that moment did not appear to be conceivable outside of the framework of the only "spaces" considered worthy of attention by analysts - that is to say the so-called "topological" spaces (in which the notion of continuous variation is applicable). One can say that the new geometry is, above all else, a synthesis between these two worlds, which, though next-door neighbors and in close solidarity, were deemed separate: the arithmetical world, wherein one finds the (so-called) spaces without continuity, and the world of continuous magnitudes, "spaces" in the conventional meaning of the word. In this new vision these two worlds, formerly separate, comprise but a single unity.
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(*) (For the mathematical reader) The "constructions and arguments" we are referring to are associated with the Cohomology of differentiable and complex varieties, in particular those which imply the Lefschetz fixed point theorems and Hodge Theory.
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The embryonic vision of this Arithmetical Geometry" (as I propose to designate the new geometry) is to be found in the Weil conjectures. In the development of some of my principal ideas (**) these conjectures were my primary source of inspiration, all through the years between 1958 and 1969.
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(**)I refer to four "intermediate" themes (nos. 5 to 8) that is to say, the topos, étale and l-adic cohomology, motives and (to a lesser extent) crystals. These themes were all developed between 1958 and 1966
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Even before me, in fact, Oscar Zariski on the one hand and Jean-Pierre Serreon the other had developed, for certain "wild" spaces in "abstract" Algebraic Geometry, some "topological" methods, inspired by those which had formerly been applied to the "well behaved spaces" of normal practice.(***)
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(***) (For the mathematical reader) The primary contribution of Zariski in this sense seems to me to be the introduction of the "Zariski topology" (which later became an essential tool for Serre in FAC), his "principle of connectedness", and what he named
the "theory of holomorphic functions" - which in his hands became the theory of formal schemes, and the theorems comparing the formal to the algebraic (with, as a secondary source of inspiration, the fundamental article by Serre known as GAGA). As for the contribution by Serre to which I've alluded in the text, it is, above all, his introduction into abstract Algebraic Geometry of the methodology of sheaves, in FAC (Faisceaux algébriques cohérents) the other fundamental paper already mentioned. In the light of these 'reminiscences", when asked to name the immediate "ancestors" of the new geometric vision, the names that come to me right away are Oscar Zariski, André Weil, Jean Leray and Jean-Pierre Serre.
Serre had a special role apart from all the others because of the fact that it was largely through him that I not only learned of his ideas, but also those of Zariski, Weil and Leray which were to play an important role in the emergence and development of the ideas of the new geometry.
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Their ideas, without a doubt, had played an important part from my very first steps towards the building of the new geometry: furthermore, it's true, as points of departure and as tools (which I had to reshape virtually from scratch in order to adapt them to a larger context), and a sources of inspiration which would continue to nourish my projects and dreams over the course of months and years. In any case, it's self-evident that, even in their recast state, these tools were insufficient for what was needed in making even the first steps in the direction of Weil's marvelous conjectures. --------------------------------------------------------------------------------
11. The Magical Spectrum - or Innocence
The two powerful ideas that had the most to contribute to the initiation and development of the new geometry are schemes and toposes, having made their appearance in a somewhat symbiotic fashion at more or less the same time.
The concept of a locale or of a "Grothendieck topology" (a preliminary form of the topos) can clearly be discerned in the wake of the scheme. This, in its turn, supplies the needed new language for ideas such as "descent" and "localization", which are employed at every stage in the development of this theme and of the schematic tools. The more inherently geometric notion of the topos, which one found only implicitely
in the work of the following years, really began to define itself clearly from about 1963, with the development of étale cohomology. Bit-by-bit however it took its rightful place as the more fundamental of the two notions. To conclude this guided tour around my opus, I still need to say a few more words about these two principal ideas.
The concept of the scheme is the natural one to start with. As "self-evident" as one could imagine, it comprises in a single concept an infinite series of versions of the idea of an (algebraic) variety, that were previously used (one version for each prime number (*).
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(*)It is convenient to include as well the case p = "infinity", corresponding to algebraic varieties of "nul characteristic".
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In addition, one and the same "scheme" (or "variety" in the new sense) can give birth, for each prime number p, to a well-defined "algebraic variety of characteristic p". The collection of these different varieties with different characteristics can thereby be seen as a kind of" (infinite) spectrum of varieties", (one for each characteristic). The "scheme" is in fact this magical spectrum, which connects between them, as so many different "branches", its "avatars", or "incarnations" in all possible characteristics. By virtue of this it furnishes an effective "principle of transition" for tying together these "varieties", arising out of geometries which, up until that point, seemed more or less isolated, cut off from each other. For the present they are all ensconced within a common "geometry" that establishes the connections between them. One might call it Schematic Geometry, the first draft of the "Arithmetic Geometry", which was able to blossom in the coming years.
The very notion of a scheme has a childlike simplicity- so simple, so humble in fact that no one before me had the audacity to take it seriously. So "infantile" in fact, that for many years afterwards, and in spite of all the evidence, for so many of my "learned" colleagues, it was treated as a triviality. In fact I needed several months of lonely investigation to fully convince myself that the idea really "worked" - that this new language,(which, however infantile it might appear, I, in my incurable naiveté continued to insist upon as something to be tested) was quite adequate for the understanding of, in a new light, with increased subtlety and in a general setting, some of the most basic geometric intuitions associated with these "geometries of characteristic p". It was a kind of exercise, prejudged by every "well informed" colleague as something idiotic and had the imagination to propose, and even (nurtured by my private demon...) follow through against all opposition!
Rather than allowing myself to be deterred by the consensus that had laid down the law over what was to be "taken seriously", and what was not, my faith was invested (as it had been in the pas) in the humble voice of phenomena, and that faculty in me which knew how to listen to it. My reward was immediate and above all expectation. In the space of only a few months, without intending to do so, I'd put my finger on several unanticipated yet very powerful tools. They've allowed me, not only to recast (as if it
were play) some old results deemed difficult, in a penetrating light that went far beyond them, but also to approach and solve certain problems in "geometries of characteristic p" that until that moment had appeared inaccessible through all known methods.(*)
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(*)The "proceedings" of this "forced inauguration" of the theory of schemes was the topic of my lecture at the International Congress of Mathematicians at Edinborough in 1958. The text of that talk would seem to me to be one of the best introductions to the subject from the aspect of schemes, and such as to perhaps influence a geometrician who reads it to make himself familiar, for better or worse, with the formidable treatise that followed it : Elements of Algebraic Geometry ("Eléments de Géométrie Algébrique" ), which treats in a detailed (without going into technicalities!), the new foundations and the new techniques of Algebraic Geometry.
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In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more or less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.
This unique power is in no way a privilege given to "exceptional talents" - persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so endowed at birth," far beyond the ordinary".
Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the "invisible yet formidable boundaries” that encircle our universe. Only innocence can surmount them, which mere knowledge doesn't even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child play.
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12: Topology - or how to survey the fogs
The innovative notion of the "scheme", we've seen, allows one to establish connections between the different geometries associated with each prime number (or "characteristics"). These geometries, however, are all of an essentially 'discontinuous'
or 'discrete' nature, as opposed to the traditional geometry which is our legacy from previous centuries, 9 back to Euclid). The new concepts introduced by Zariski and by Serre have restored, to some extent, a 'continuous dimension' concept for these geometries, which was automatically picked up by the "schematic geometry" that had just been invented to unify them. However the "fabulous conjectures" of Weil were still a long way off. These "Zariski topologies" were, seen from this perspective, so crude that one might just as well have remained at the "discrete aggregate" stage.
It was clear that what was still lacking was some new principle that could connect these geometric objects (or "varieties", or "schemes") to the usual "well behaved" (topological) "spaces": those, let us say, whose points are clearly distinguished one from the other, whereas in the "harum-scarum" spaces introduced by Zariski, the points have a sneaky tendency to cling to one another....
Most certainly it was through nothing less than through this "new principle" that the marriage of "number and magnitude", (or of "continuous and discontinuous" geometry) could give birth to the Weil conjectures.
The notion of space is certainly one of the oldest in mathematics. It is fundamental to our "geometric" perspective on the world, and has been so tacitly for over two millenia. Its only over the course of the 19th century that this concept has, bit-by-bit, freed itself from the tyranny of our immediate perceptions (that is, one and the same as the "space" that surrounds us), and of its traditional theoretical treatment (Euclidean), to attain to its present dynamism and autonomy. In our own times it has joined the ranks of those notions that are most freely and universally employed in mathematics, and is familiar, I would say, to every mathematician without exception. It has become a concept of multiple and varied aspects, of hundreds of thousands of faces depending on the kinds of structures one chooses to impose on a space, from the most abundant and rich, (such as the venerable 'Euclidean' structures, or the 'affine' or 'projective' ones, or again the 'algebraic' structures of similarly designated 'varieties' which generalize and extend them), down to the most 'impoverished': those in which all 'quantitative' information has been removed without a trace, or in which only a qualitative essence of "proximity" or of "limit"(*), (and, in its most elusive version, the intuition of form (called 'topological spaces' )), remains.
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(*)When I speak of the idea of a "limit" it is above all in terms of passage to a limit, rather than the idea that most non-mathematicians, of a "frontier"
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The most "reductive" of all these notions over the course of half a century down to the present, has appropriated to itself the role of a kind of conceptual englobing substrate for all the others, that of the topological space. The study of these spaces constitutes one of the most fascinating and vital branches of geometry: Topology.
As elusive as it might appear initially, the "qualitatively pure" structure encapsulated in the notion of "space"(topological) in the absence of all quantitative givens, (notably the metric distances between points) which enables us to relate it to habitual
intuitions of "large" and "small", we have, all the same, over the last century, been able to confine these spaces in the locked flexible suitcases of a language which has been meticulously fabricated as the occasion arose. Still better, as the occasion arose, various 'weights and measures' have been devised to serve a general function, good or bad, of attaching "measures" (called 'topological invariants'), to those sprawled-out spaces which appear to resist, like fleeting mists, any sort of metrizability. Most of these invariants, its true, certainly the most essential ones, are more subtle than simple notions like 'number' and 'magnitude' - often they are themselves fairly delicate mathematical structures bound (by rather sophisticated constructions) to the space in question. One of the oldest and most crucial of these invariants, introduced in the last century (by the Italian mathematician Betti) is formed from the various "groups" (or 'spaces'), called the "Cohomology" associated with this space. (*) --------------------------------------------------------------------------------
(*) Properly speaking, the Betti invariants were homological invariants. Cohomology is a more or less equivalent or "dual" version that was introduced much later. This has gained pre-eminence over the initial "homological" aspect, doubtless as a consequence of the introduction, by Jean Leray, of the viewpoint of sheaves, which is discussed further on. From the technical point of view one can say that a good part of my work in geometry has been to identify and develop at some length, the cohomological theories which were needed for spaces and varieties of every sort, above all for the "algebraic varieties" and the schemes. Along the way I was also led to a reinterpretation of the traditional homological invariants in cohomological terms, and through doing so, to reveal them in an entirely new light.
There are numerous other "topological invariants" which have been introduced by the topologists to deal with this or that property or this or that topological space. Next after the "dimension" of a space and the (co)homological invariants, come the "homotopy groups". In 1957 I introduced yet another one, the group (known as "Grothendieck) K(X), which has known a sensational success and whose importance (both in topology and arithmetic) is constantly being re-affirmed. A whole slew of new invariants, more sophisticated than the ones presently known and in use, yet which I believe to be fundamental, have been predicted by my "moderated topology" program (one can find a very summary sketch of this in the "Outline for a Program" which appears in Volume 4 of the Mathematical Reflections). This program bases itself on the notion of a "moderated theory" or "moderated space", which constitutes, a bit like the topos, a second "metamorphosis of the concept of space". It is at the same time more self-evident and less profound than the latter. I predict that its immediate applications to topology "properly speaking" will be decidedly more incisive, that in fact it will turn upside down the "profession" of topological geometer, through a far-reaching transformation of the conceptual context appropriate to it. (As was the case with Algebraic Geometry with the introduction of the point-of-view of the scheme) Furthermore, I've already sent copies of my "Outline" to several of my old friends and some illustrious topologists, yet it seems to me that that haven't been inclined to take any interest in it....
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It was the Betti numbers that figure ("between the lines" naturally) in the Weil conjectures, which are their fundamental "reason for being" and which (at least for me, having been "let in on the secret" by Serre's explications) give them meaning. Yet the possibility of associating these invariants with the "abstract" algebraic varieties that enter into these conjectures, in such a manner as to response to the very precise desiderata demanded by the requirements of this particular cause - that was something only to be hoped for. I doubt very much that, outside of Serre and myself, there isn't anyone else (including André Weil!) who really believes in it.(*)
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(*) It is somewhat paradoxical that Weil should have an obstinate, even visceral block against the formalism of cohomology, particularly since it had been in large part his "famous" conjectures that inspired the development, starting in 1955, of the great cohomological theories of algebraic geometry, (launched by J.P. Serre with his foundational article "FAC", already alluded to in a footnote.)
It’s my opinion that this "block" is part of a general aversion in Weil against all the global formalisms, (whether large or small), or any sort of theoretical construction. He hasn't anything of the true "builder" about him, and it was entirely contrary to his personal style that he saw himself constrained to develop, starting with the 30's, the fundamentals of "abstract" algebraic geometry, which to him (by his own dispositions), have proved to be a veritable "Procrustean bed" for those who use them.
I hope he doesn't hold it against me that I chose to go beyond him, investing my energy in the construction of enormous dwelling places, which have allowed the dreams of a Kronecker, and even of himself, to be cast into a language and tools that are at the same time effective and sophisticated. At no time did he ever comment to me about the work that he saw me doing, or which had already been done. Nor have I received any response from him about Récoltes et Semailles, which i sent to him over three months ago, with a warm hand-written personal dedication to him.
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Soon afterwards our understanding of these cohomological invariants was profoundly enriched and renovated by the work of Jean Leray (carried out as a prisoner of war in Germany in the early part of the 40's). The essential novelty in his ideas was that of the (Abelian) sheaf over a space, to which Leray associated a corresponding collection of cohomology groups (called "sheaf coefficients"). It is as if the good old standard "cohomological metric" which had been used up to them to "measure" a space, had suddenly multiplied into an unimaginablely large number of new "metrics" of every shape, size and form imaginable, each intimately adapted to the space in question, each supplying us with very precise information which it alone can provide. This was the dominant concept involved in the profound transformation of our approach to spaces of every sort, and unquestionably one of the most important mathematical ideas of the 20th century.
Thanks above all to the ulterior work of Jean-Pierre Serre, Leray's ideas have produced in the half century since their formulation, a major redirection of the whole theory of topological spaces, (notably those invariants designated as "homotopic",
which are intimately allied with cohomology), and a further redirection, no less significant, of so-called "abstract" algebraic geometry (starting with the FAC article of Serre in 1955). My early work in geometry, from 1955 onwards, was conceived of as a continuation of the work of Serre, and for that reason also a continuation of the work of Leray.
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13: Toposes - or the Double Bed
The new perspective and language introduced by the use of Leray's concepts of sheaves has led us to consider every kind of "space" and "variety" in a new light. These did not however have anything to say about the concept of space itself, and was content if it enabled us to refine our understanding of the already traditional and familiar "spaces". At the same time it was recognized that this way of looking at space was insufficient for taking into account the "topological invariants" which were most essential for expression the "form" of these "abstract algebraic varieties" (such as those which figure in the Weil Conjectures), let alone that of general "schemes" (for the most part the classical varieties). For the desired "marriage" of "Number and Magnitude" one would have a rather narrow bed, one in which at most one of the future spouses (for example, the bride) could accommodate herself for better or worse, but never both at the same time! The "new principle" that needed to be found so that the marriage announced by the guardian spirits could be consummated, was simply that missing spacious bed, though nobody at the time suspected it.
This "double bed" arrived (as from the wave of a magic wand) with the idea of the topos. This idea encapsulates, in a single topological intuition, both the traditional topological spaces, incarnation of the world of the continuous quantity, and the so-called "spaces" (or "varieties") of the unrepentant abstract algebraic geometers, and a huge number of other sorts of structures which until that moment had appeared to belong irrevocably to the "arithmetic world" of "discontinuous" or "discrete"
aggregates.
It was certainly the sheaf perspective that was my sure and quiet guide, the right key (hardly secret) to lead me without detours or procrastination towards the nuptial chamber and its vast conjugal bed. A bed so enormous in fact (like a vast, deep and peaceful stream) in which
"Tous Les Chevaux du RoiY pourraient boire ensemble"- as the old ballad that you must surely have heard or sung at one point tells us. And he who was the first to sing it was he who has best savored the secret beauty and passive force of the topos, better than any of my clever students and former friends....
It was the same key, both in the initial and provisional approach via the convenient, yet unintrinsic, concept of a "site", as with the topos. I will now attempt to describe the topos concept.
Consider the set formed by all sheaves over a (given) topological space or, if you like, the formidable arsenal of all the "rulers" that can be used in taking measurements on it. (*)
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(*)(For the mathematician): properly speaking, one is speaking of sheaves of ensembles, not the Abelian sheaves introduced by Leray as generalized coefficients in the formation of "cohomology groups" I believe that I'm the first person to have worked systematically with sheaves of ensembles (starting in 1955 at the University of Kansas, with my article "A general theory of fibre spaces with structure sheaf")
We will treat this "ensemble", or "arsenal" as one equipped with a structure that may be considered "self-evident", one that crops up "in front of one's nose": that is to say, a Categorical structure. (Let not the non-mathematical reader trouble himself if he's unaware of the technical meanings of these terms, which will not be needed for what follows).
It functions as a kind of "superstructure of measurement", called the "Category of Sheaves" (over the given space), which henceforth shall be taken to incorporate all that is most essential about that space. This is in all respects a lawful procedure, (in terms of "mathematical common sense") because it turns out that one can "reconstitute" in all respects, the topological space(**) by means of the associated "category of sheaves" (or "arsenal" of measuring instruments)
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(**) (For the mathematical reader) Strictly speaking, this is only true for so-called "tame" spaces. However these include virtually all of the spaces one has to deal with, notably the "separable spaces" so dear to functional analysts.
(The verification of this is a simple exercise- once someone thinks to pose the question, naturally) One needs nothing more (if one feels the need for one reason or another), henceforth one can drop the initial space and only hold onto its associated "category"
(or its "arsenal"), which ought to be considered as the most complete incarnation of the "topological (or spatial) structure" which it exemplifies
As is often the case in mathematics, we've succeeded (thanks to the crucial notion of a "sheaf" or "cohomological ruler") to express a certain idea (that of a "space" in this instance), in terms of another one (that of the "category"). Each time the discovery of such a translation from one notion (representing one kind of situation) to another (which corresponds to a different situation) enriches our understanding of both notions, owing to the unanticipated confluence of specific intuitions which relate first to one then to the other. Thus we see that a situation said to have a "topological" character (embodied in some given space) has been translated into a situation whose character is "algebraic" (embodied in the category); or, if you wish, "continuity" (as present in the space) finds itself "translated" or "expressed" by a categorical structure of an "algebraic" character, (which until then had been understood only in terms of something "discrete" or "discontinuous".)
Yet there is more here. The first idea, that of the space, was perceived by us as a "maximal" thing - a notion already so general that one could hardly envisage any kind of "rational" extension to it. On the contrary, it has turned out that, on the other side of the mirror (*)
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(*) The "mirror" referred to, as in Alice in Wonderland, is that which yields as the "image" of a space placed in front of it, the associated "category", considered as a kind of "double" of the space, on the "other side of the mirror(*) these "categories", (or "arsenals") one ends up with in dealing with topological spaces, are of a very particular character. Their collection of traits is in fact highly specific.(**), and tend to join up in patchwork combinations of an unbelievably simple nature- those which on can obtain by taking as one's point of departure the reduction of a space to a single point.
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(**) (For the mathematical reader) We're speaking about primarily the properties which I introduced into Category Theory under the name of "exact characteristics", (along with the categorical notions of general projective and inductive limits). See "On several points of homological algebra", Tohoku Math Journal, 1957 (p. 119-221))
Having said this, a "space defined in the new way" (or topos) one that generalizes the traditional topological space, can be simply described as a "category" which, without necessarily deriving from an ordinary space, nevertheless possesses all of the good properties (explicitly designated once and for all, naturally) of the "sheaf category". --------------------------------------------------------------------------------
This therefore is the new idea. Its appearance may perhaps be understood in the light of the observation, a childlike one at that, that what really counts in a topological space is neither its "points" nor its subsets of points.(*), nor the proximity relations between them, rather it is the sheaves on that space, and the category that they produce.
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(*)Thus, one can actually construct "enormous" topoi with only a single point, or without any points at all!
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All that I've done was to draw out the ultimate consequences of the initial notion of Leray - and by doing so, lead the way. As even the idea of sheaves (due to Leray), or that of schemes, as with all grand ideas that overthrow the established vision of things, the idea of the topos had everything one could hope to cause a disturbance, primarily through its "self-evident" naturalness, through its simplicity (at the limit naive, simp le-minded, "infantile")- through that special quality which so often makes uscry out: "Oh, that's all there is to it!", in a tone mixing betrayal with envy, that innuendo of the "extravagant", the "frivolous", that one reserves for all things that are unsettling by their unforseen simplicity, causing us to recall, perhaps, the long buried days of our infancy....
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14. Mutability of the Concept of Space-or Breath and Faith
The notion of the scheme constitutes a great enrichment of the notion of the "algebraic variety". By virtue of that fact it has successful renovated, from top to bottom, the subject of Algebraic Geometry left to me by my predecessors. The notion of the topos however constitutes an altogether unsuspected extension, more accurately a metamorphoses of the concept of space. Thereby it holds the promise to effect a similar renovation of the subject of Topology and, beyond that, Geometry. Furthermore, at present it has already played a crucial role in the growth and development of the new geometry (above all by means of the methods of p-adic and crystalline cohomology which have come out of it and, thereby, the proofs of the Weil conjectures.) As its elder sister (quasi twin) it contains the pair of complementary characteristics essential to every fertile generalization, to wit: Primo, the new concept isn't too large, in the sense that within these new "spaces", (or, for the sake of overly delicate ears [1], "toposes") the most essential "geometric" intuitions [2] and constructions, familiar to us from the old traditional spaces, can be easily transposed in an evident manner. In other words, one has at one's disposal in these new objects the rich collection of images and mental associations, of ideas and certainly some techniques, that were formerly confined to objects of the earlier sort.
Secundo, the new concept is large enough to encapsulate a host of situations which, until now, were not considered capable of supporting intuitions of a "topologic-geometric" nature - those intuitions, indeed, which had been reserved in the past exclusively for the ordinary topological spaces (and for good reason....)
What is crucial, from the standpoint of the Weil conjectures, is that the new ideas be ample enough to allow us to associate with every scheme such a "generalized space" or "topos" (called the "étale topos" of the corresponding scheme). Certain "cohomological invariants" of this topos (nothing can be more "childishly simple"!) then appeared to furnish one with "what was needed" in order to bring out the full meaning of these conjectures, and perhaps (who knew then!) supply the means for demonstrating them.
It's in the pages that I'm in the process of writing at this very moment that, for the first time in my life as a mathematician, I can take the time needed to evoke (if only for myself) the ensemble of the master-themes and motivating ideas of my mathematical work. It's lead me to an appreciation of the role and the extensions of each of these themes and the "viewpoints" they incarnate, in the great geometric vision that unite them and from which they've issued. It is through this work that the two innovative ideas of the first powerful surge of the new geometry first saw the light of day: that of schemes and that of the topos.
It's the second of these ideas, that of the topos, which at this moment impresses me as the more profound of the two of them. Given that I, at the end of the 50's, rolled up my sleeves to do the obstinate work of developing, through twelve long years, of a "schematic tool" of extraordinary power and delicacy, it is almost incomprehensible to me that in the ten or twenty years that have since followed, others besides myself have not carried through the obvious implications of these ideas, or raised up at least a few dilapidated "prefabricated" shacks as a contribution to the spacious and comfortable mansions that I had the heart to build up brick by brick and with my own bare hands.
At the same time, I haven't seen anyone else on the mathematical scene, over the last three decades, who possesses that quality of naiveté, or innocence, to take (in my place) that crucial step, the introduction of the virtually infantile notion of the topos, (or even that of the "site"). And, granted that this idea had already been introduced by myself, and with it the timid promise that it appeared to hold out - I know of no-one else, whether among my former friends or among my students, who would have had the "wind", and above all the "faith", to carry this lowly notion [3] to term (so insignificant at first sight, given that the ultimate goal appeared infinitely distant...) : since its first stumbling steps, all the way to full maturity of the "mastery of étale cohomology", which, in my hands, it came to incarnate over the years that followed.
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[1] Nomenclature: the name "topos" was chosen (with its associations to "topology"
and "topological"), to imply that it was the "principal object" to which "topological intuition" inheres. Through the rich cloud of mental images that this name evokes, one ought to consider it as more or less equivalent to the term "space" (topological), with the requirement that the notion of the "topological" be more precisely specified. (In the same way that one has "vectorial spaces", but on "vectorial toposes", at least for the moment!) It's important to maintain both expressions together, each with its proper specificity.
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[2] Among these "constructions" one finds the familiar "topological invariants", including the cohomological invariants. For these I've done all that's necessary in the article previously cited ("Tohoku" 1955) in order to give them a proper meaning for each "topos".
[3] (For the mathematical reader) When I speak of "wind" and of "faith", I'm referring to characteristics of a non-technical nature, although I consider them to be essentially necessary characteristics. At another level I might add that I have referred to the "cohomological flair", that is to say the sort of aptitude that was developed in me through the erection of theories of cohomology. I believed that I was able to transmit this to my students in cohomology. With a perspective of 17 years after my departure from the world of mathematics, I can say that not a one of them had developed it. --------------------------------------------------------------------------------
15: Tous les Chevaux du Roi...
Verily, the river is deep, and peaceful and vast are the waters of my infancy, in a kingdom which I'd believed to have left so long ago. All the king's horses may come and drink at their leisure, quenching of their thirst without the waters ever drying up! They descend from the glaciers, full of the ardor of distant snows, with the sweetness of the clay of the plains. I've just written about one of those horses, which were led to drink by a child and which drank at length to its full content. And I saw another that came to drink for a moment or two, in search of that same youngster - but it did not linger. Someone must have chased it off. And, to speak truly, that's all. Yet I also see numberless herds of horses who wander the plains, dying of thirst - as recently as this morning their whinnying dragged me from my bed, and at an unaccustomed hour, although I am on the verge of my 60's and cherish my tranquility. There was no help for it; I was obliged to get up. It gave me pain to see them, horridly raw-boned and skinny, although there was no lack of abundance of good water or green pasture. Yet one might speak of a kind of malignant magical spell that has fallen over the land that I once found so accommodating, contaminating its generous waters. Who knows? One
could imagine that some kind of plot had been hatched by the horse-traders of the land to bring down prices! Or it may be that this country no longer possesses any children for leading the horses to water, and that the horses will remain thirsty until there is a child who rediscovers the road that leads to the stream....
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16: Motives, or the Inner heart
The "topos" theme came from that of "schemes" in the year of their appearance; yet it has greatly surpassed the mother notion in its extent. It is the topos, not schemes, which is the "bed", or that "deep river", in which the marriage of geometry, topology and arithmetic, mathematical logic, the theory of categories, and that of continuous and discontinuous or "discrete" structures, is celebrated. If the theme of schemes is at the heart of the new geometry, the theme of the topos envelopes it as a kind of residence. It is my grandest conception, devised in order to grasp with precision, in the same language rich in resonances of geometry, an "essence" common to the most disparate situations, coming from every region of the universe of mathematical objects.
Yet the topos has not known the good fortune of the schemes. I discuss this subject in several places in Récoltes et Semailles, and this is not the place at which to dwell upon the strange adventures which have befallen this concept. However, two of the principal themes of the new geometry have derived from that of the topos, two "cohomological theories" have been conceived, one after the other, with the same purpose of providing an approach to the Weil conjectures: the étale (or l-adic) theme, and the crystalline theme.
The first was given concrete form in my hands as the tool of l-adic cohomology, which has been shown to be one of the most powerful mathematical tools of this century.
As for the crystalline theme, (which had been reduced since my departure to a virtually quasi-occult standing), it has finally been revitalized (under the pressure of necessity), in the footlights and under a borrowed name, in circumstances even more bizarre than those which have surrounded the topos.
As predicted, it was the tool of l-adic cohomology which was needed to solve the Weil conjectures. I did most of the work, before the remainder was accomplished, in a
magistral fashion, 3 years after my departure, by Pierre Deligne, the most brilliant of all my "cohomological" students.
Around 1968 I came up with a stronger version, (more geometric above all), ofthe Weil conjectures. These are still "stained" (if one may use that expression) with an "arithmetical" quality which appears to be irreducible. All the same, the spirit of these conjectures is to grasp and express the "arithmetical" (or discrete) through the mediation of the "geometric" (or the "continuous".)(*)
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(*For the mathematical reader) The Weil conjectures are subject to hypotheses of an essentially arithmetical nature, principally because the varieties involved must be defined over finite fields. From the point of view of the cohomological formalism, this results in a privileged status being ascribed to the Frobenius endomorphism allied with such situations. In my approach, the crucial properties (analogous to 'generalized index theorems') are present in the various algebraic correspondences, without making any arithmetic hypotheses about some previously assigned field.
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In this sense the version of these conjectures which I've extracted from them appears to my mind to be more "faithful" to the "Weil philosophy" than those of Weil himself! - a philosophy that has never been written down and rarely expressed, yet which probably has been the primary motivating force in the extraordinary growth and development of geometry over the course of the last 4 decades.(*)
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(*)Since my departure in 1970 however, a reactionary tendency has set in, finding its concrete expression in a state of relative stagnation, which I speak of on several occasions in the pages of Récoltes et Semailles.
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My reformulation consisted, essentially, in extracting a sort of "quintessence" of what is truly valuable in the framework of what are called "abstract" algebraic varieties, in classical "Hodge theory", and in the study of "ordinary" algebraic varieties.(*) --------------------------------------------------------------------------------
(*)Here the word 'ordinary' signifies:"defined over complex fields". Hodge theory (for "harmonic integrals") was the most powerful of the known cohomological theories in the context of complex algebraic varieties.
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I've named this entirely geometric form of these celebrated conjectures the "standard conjectures".
To my way of thinking, this was, after the development of l-adic cohomology, a new step in the direction of these conjectures. Yet, at the same time and above all, it was also one of the principal possible approaches towards what still appears to me to be the most profound of all the themes I've introduced into mathematics (*), that of motives, (themselves originating in the "l-adic cohomology theme")
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(*)This was the deepest theme at least during my period of mathematical activity
between 1950 and 1969, that is to say up to the very moment of my departure from the mathematical scene. I deem the themes of an abelian algebraic geometry and that of Galois-Teichmuller theory, which have developed since 1977, to be of comparable depth.
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This theme is like the heart, or soul, that which is most hidden, most completely shielded from view within the "schematic" theme, which is itself at the very heart of the new vision. And several key phenomena retrieved from the standard conjectures (**) can also be seen as constituting a sort of ultimate quintessence of the motivic theme, like the "vital breath" of this most subtle of all themes, of this "heart within the heart" of the new geometry.
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(**) (For the algebraic geometer). Sooner or later there must be a revision of these conjectures. For more detailed commentary, go to "The tower of scaffoldings" (R&S IV footnote #178, p. 1215-1216), and the note at the bottom of page 769, in "Conviction and knowledge" (R&S III, footnote#162)
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Roughly speaking, this is what's involved. We've come to understand, for a given prime number p, the importance of knowing how to construct "cohomological theories" (particularly in light of the Weil conjectures) for the "algebraic varieties of characteristic p". Now, the celebrated "cohomological l-adic tool" supplies one with just such a theory, and indeed, an infinitude of different cohomological theories, that is to say, one associated with each prime number different from p. Clearly there is a "missing" theory, namely that in which l and p are equal. In order to provide for this case I conceived of yet another cohomological theory (to which I've already alluded), entitled "crystalline cohomology". Furthermore, in the case in which p is infinite, there are yet 3 more cohomological theories (***)
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(***) (For the benefit of the mathematical reader) These theories correspond, respectively, to Betti cohomology(by means of transcendental , and with the help of an embedding of the base field into the field of the complex numbers), Hodge cohomology, and de Rham cohomology as interpreted by myself. The latter two date back to the 50's (that of Betti to the 19th century).
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Furthermore there is nothing to prevent the appearance, sooner or later, of yet more cohomological theories, with totally analogous formal properties. In contradistinction to what one finds in ordinary topology, one finds oneself in the presence of a disconcerting abundance of differing cohomological theories. One had the impression that, in a sense that should be taken rather flexibly, all of these theories "boiled down" to the same one, that they "gave the same results". (****)
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(****)(For the benefit of the mathematical reader). For example, if f is an endomorphism of the algebraic variety X, inducing an endomorphism of the cohomology space Hi(X), then the fact that the "characteristic polynomial" of the latter
must have integrer coefficients does not depend on the kind of cohomology employed (for example, l-adic for some arbitrary l). Likewise for algebraic correspondences in general, which X is presumed proper and smooth. The sad truth, (and this gives one an idea of the deporable state in which the cohomological theory of algebraic varieties of characterstic p finds itself since my departure), is that there is no demonstration of this fact, as of this writing, even in the simplest case in which X is a smooth projective surface, and i =2. Indeed, to my knowledge, nobody since my departure has deigned to interest himself in this crucial question, which is typical of all those which are subsidiary to the standard conjecture. The doctrine a-la-mode is that the only endomorphism worthy of anyone's attention is the Frobenius endomorphism, (which could have been treated by Deligne by the method of boundaries...)
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It was through my intention to give expression to this "kinship" between differing cohomological theories that I arrived at the notion of associating an algebraic variety with a "motive". My intention in using this term is to suggest the notion of the "common mo
tive" (or of the "common rationale") subsidiary to the great diversity of cohomological invariants associated with the variety, owing to the enormous collection of cohomologies possible apriori. The differing cohomological theories would then be merely so many differing thematic developments, (each in the "tempo", the "key", and "mode" ("major" or "minor")appropriate to it), of an identical "basic motive" (called the "motivic cohomological theory" ), which would also be at the same time the most fundamental, the ultimate "refinement" of all the differing thematic incarnations (that is to say, of all the possible cohomological theories).
Thus the motive associated with an algebraic variety would constitute the ultimate invariant, the invariant par excellence from the cohomological standpoint among so many musical "incarnations", or differing "realizations". All of the essential properties of the cohomology of the variety could already be read off (or be "extended to") on the corresponding motive, with the result that the properties and familiar structures of particular cohomological invariants, ( l-adic, crystalline for example) would be merely the faithful reflection of the properties and structures intrinsic to the motive(*). --------------------------------------------------------------------------------
(*)(For the benefit of the mathematical reader) Another way of viewing the category of motives over a field k, is to visualize it as a kind of "covering Abelian category" of the category of distinct schemes of finite type over k. Then the motive associated with a given schema X ("cohomological motive" of X which I notate as H*(mot) (X)) thereby appears as a sort of "Abelianized avatar" of X. The essential point is that, even as an Abelian variety X is susceptible to "continuous variation" (with a dependence of its' isomorphism class on "continuous parameters", or "modules"), the motive associated with X, or more generally, a "variable" motive, is also susceptible to continuous variation. This is an aspect of motivic cohomology which is in flagrant contrast to what one normally has with respect to all the classical cohomological invariants, (including the l-adic invariants), with the sole exception of the Hodge cohomology of complex algebraic varieties.
This should give one an idea of to what extent "motivic cohomology" is a more refined invariant, encapsulating in a far tighter manner the "arithmetical form" (if I can risk such an expression) of X, than do the traditional invariant
s of pure topology. In my way of looking at motives, they consitute a kind of delicate and hidden "thread" linking the algebraic-geometric properties of an algebraic variety to the properties of an "arithmetic" nature incarnated in its motive. The latter may then be considered to be an object which, in its spirit, is geometric in nature, yet for which the "arithmetic" properties implicit in its geometry have been laid bare.
Thus, the motive presents itself as the deepest "form invariant" which one has been able to associate up to the present moment with an algebraic variety, setting aside its "motivic fundamental group". For me both invariants represent the "shadows" projected by a "motivic homotopy type" which remains to be discovered (and about which I say a few things in the footnote: "The tower of scaffoldings- or tools and vision" (R&S IV, #178, see scaffolding 5 (Motives), and in particular page 1214)).
It is the latter object which appears to me to be the most perfect incarnation of the elusive intuition of "arithmetic form" (or "motivic"), of an arbitrary algebraic variety. --------------------------------------------------------------------------------
Here we find, expressed in the untechnical language of musical metaphor, the quintessence of an idea (both delicate and audacious at once), of virtually infantile simplicity. This idea was developed, on the fringes of more fundamental and urgent tasks, under the name of the "theory of motives", or of "philosophy (or "yoga") of the "motives", through the years 19673-69. It's a theory of a fascinating structural richness, a large part of which remains purely conjectural. (*)
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(*)I've explained my vision of motives to any who wished to learn about them all through the years, without taking the trouble to publish anything in black and white on this subject (not lacking in other tasks of importance). This enabled several of my students later on to pillage me all the more easily, and under the tender gaze of my circle of friends who were well aware of the situation. (See the following footnote) --------------------------------------------------------------------------------
IN R &S I often return to this topic of the "yoga of motives", of which I am particularly fond. There is no need to dwell here on what is discussed so thoroughly elsewhere. It suffices for me to say that the "standard conjectures" flow in a very natural way from the world of this yoga of motives. These conjectures furnish at the same time a primary means for effecting one of the possible formal constructions of the notion of the motive.
The standard conjectures appeared to me then, and still do today, as one of the two questions which are the most fundamental in Algebraic Geometry. Neither this question, nor the other one (known as the "resolution of singularities") has been answered at the present time. However, whereas the second of them has a venerable history of a century, the other one, which I've had the honor of discovering, now tends
to be classified according to the dictates of fad-and-fashion (over the years following my departure from the mathematical scene, (and similarly for the theme of motives) ), as some kind of genial "grothendieckean" fol-de-rol. Once more I'm getting ahead of myself.... (*)
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(*) In point of fact, this theme was exhumed (one year after the crystallinetheme), but this time under its own name, (and in a truncated form, and only in the single case of a base field of null characteristic), without the name of its discoverer being so much as mentioned. It constiutes one example amone of its discoverer being so much as mentioned. It constiutes one example among so many others, of an idea and a theme which were buried at the time of my departure as some kind of "grothendieckean" fantasmagoria", only to be revived, one after another, by certain of my students over the course of the next 10 to 15 years, with shameless pride and (need one spell it out?) never a mention of its originator.
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17. In Quest of the Mother - Two Views
Speaking truthfully, my thoughts about the Weil conjectures in and of themselves, that is to say with the goal of solving them, have been sporadic. The panorama that opened up before me, which I was obliged to make the effort to scrutinize and capture, greatly surpassed in scope and in depth the hypothetical needs for proving these conjectures, or indeed all the results that would follow from them. With the emergence of the themes of the schemes and topos, an unsuspected world suddenly opened up. Certainly the "conjectures" occupy a central place, in much the way as the capital city of a vast empire or continent, with numberless provinces, most of which have only the most tenuous relations with the brilliant and prestigious metropolis. Without having to make it explicit, I knew that henceforth I was to be the servant of a great enterprise: to explore this immense and unknown world, to depict its frontiers however far distant: to traverse it in all directions, to inventory with obstinate care the closest and most accessible of these provinces; then to draw up precise maps in which the least little village and tiniest cottage would have their proper place...
It is the later task, above all, which absorbed most of my energy - a long and patient labor on foundations, which I was the first to see with clarity and, above all, to "know
in my guts". It is this which took up the major part of my time between 1958 (the year in which one after another, the schemes and the topos made their respective appearances), and 1970, (the year of my departure from the mathematical scene.)
It often happened also that I chaffed at the bit to be constrained in this fashion, like someone pinned down by an immovable weight, by those interminable tasks which (once the essentials had been understood) seemed more of a routine character than a setting forth into the unknown. I had constantly to restrain the impulse to thrust forward - in the manner of a pioneer or explorer, occupied somewhere far distant in the discovery and exploration of unknown and nameless worlds, crying out for me to become acquainted with them and bestow names upon them. This impulse, and the energy I invested in them, (partially, in my spare time), were constantly held in abeyance.
However I knew very well that it was this energy, so slight, (in a manner of speaking)in comparison with what I gave to my "duties", that was the most important and advanced; in my "creative" work in mathematics it was this that was involved; in that intense attention given to the apprehension of, in the obscure folds, formless and moist, of a hot and inexhaustibly nourishing womb, the earliest traces and shapes of what had yet to be born and which appeared to be calling out to me to give it form, incarnation and birth... This work of discovery, the concentrated attention involved, and its ardent solicitude, constituted a primeval force, analogous to the sun's heat in the germination and gestation of seeds sown in the nourishing earth, and for their miraculous bursting forth into the light of day.
In my work as a mathematician I've seen two primary forces or tendencies of equal importance at work, yet of totally different natures- or so it seems to me. To evoke them I've made use of the images of the builder, and of the pioneer or explorer. Put alongside each other, both strike me somehow as really quite "yang", very "masculine", even "macho"! They possess the heightened resonance of mythology, of "great events". Undoubtably they've been inspired by the vestiges within me of my old "heroic" vision of the creative worker, the "super-yang" vision. Be that as it may, they produce a highly colored image, if not totally pictorial yet "standing at attention" to be viewed, of a far more fluid, humble and "simple" reality -one that is truly living.< P> However, in this "male" "builder's" drive, which would seem to push me relentlessly to engineer new constructions I have, at the same time, discerned in me something of the homebody, someone with a profound attachment to " the home". Above all else, it is "his" home, that of persons "closest" to him- the site of an intimate living entity of which he feels himself a part. Only then, and to the degree which the circle of his "close associates" can be enlarged, can it also be an "open house" for everyone.
And, in this drive to "make" houses (as one "makes" love...) there is above all, tenderness. There is furthermore the urge for contact with those materials that one shapes a bit at a time, with loving care, and which one only knows through that loving
contact. Then, once the walls have been erected, pillars and roof put in place, there comes the intense satisfaction of installing the rooms, one after the other, and witnessing the emergence, little by little, from these halls, rooms and alcoves, of the harmonious order of a living habitation]n - charming, welcoming, good to live in. Because the home, above all and secretly in all of us, is the Mother - that which surrounds and shelters us, source at once of refuge and comfort; and it is even (at a still deeper level, and even as we are in the process of putting it all in place), that place from which we are all issued, which has housed and nourished us in that unforgettable time before our birth... It is thus also the Busom.
And the other spontaneously generated image, going beyond the inflated notion of a "pioneer', and in order to grasp the hidden reality which it conceals, is itself devoid of all sense of the "heroic". There once again, it is the archetypal maternal image which occurs - that of the nourishing "matrix” and of its formless and obscure labors...
These twin urges which appeared to me as being "totally different" have turned out to be much closer than I would have imagined. Both the one and the other have the character of a "drive for contact", carrying us to the encounter with "the Mother": that which incarnates both that which is close and "known", and that which is "unknown". In abandoning myself to either one or the other, it is to "rediscover the Mother", it is in order to renovate contact with that which is near, and "more or less known", and that which is distant, yet at the same time sensed as being on the verge of being understood.
The distinction is primarily one of tone, of quantity, but not of an essential nature. When I "construct houses", it is the "known" which dominates; when I "explore", it is the "unknown". These two "modes" of discovery, or to better state the matter, these two aspects of a single process, are indissolubly linked. Each is essential and complementary to the other. In my mathematical work I've discerned a coming-and-going between these two ways of approaching things, or rather, between those moments (or periods) in which one predominates, then the other (*)
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(*) What I've been saying about mathematical work is equally true for "meditative" activity (which is discussed more or less throughout Récoltes et Semailles). I have no doubts that it is innate to all forms of discovery, including those of the artist (writer or poet for example). The two "faces" which I've described here might also be seen as being, on the one hand that of expression and its "technical" requirements, while the other is that of reception (of perceptions and impressions of all sorts), turning into inspiration as a consequence of intense concentration. Both the one and the other are present at every working moment, as well as that 'coming-and-going', in which first the one predominates, then the other.
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Yet it is also clear that, at every instant, one or the other mode will be present. When I construct, furnish, clear out the rubble or clean the premises, or set things in order, it
is the "mode", or "face" of the "yang", the "masculine" which sets the tone of my work. When I explore, groping around that which is uncomprehended, formless, that which is yet without any name, I'm following the "yin" aspect, or "feminine" side of my being. I've no intention of wishing to minimize or denigrate either side of my nature, each essential one to the other: the "masculine" which builds and engenders, or the "feminine" which conceives, which shelters the long and obscure pregnancies. I "am" either one or the other - "yang" and "yin", "man" and "woman". Yet I'm also aware that the more delicate, the subtler in unraveling of the secretive processes is to be found in the "yin" or "feminine" aspect - humble, obscure, often mediocre in appearance.
It's this side of my labor which, always I would say, has held the greatest fascination for me. The modern consensus however had tried to encourage me to invest the better part of my energy in the other side, in those efforts which affirm themselves by being incarnated in "tangible" products, if not always finished or perfected - products with well-defined boundaries, asserting their reality as if they'd been cut in stone... I can now see, upon reflection, how heavily this consensus weighed on me, and also how I "bore the weight of the accusation"-with submission! The aspect of "conception” or "exploration" of my work was accorded a meager role by me, even up to the moment of my departure. And yet, in the retrospective overview I've made of my work as a mathematician, the evidence leaps out to me that the thing that has constituted the very essence and power of this work, has been the face which, in today's world, is the most neglected, when it is not frankly treated as an object of derision or disdainful condescension: that of the ideas, even that of dreams, never that of results.
In attempting in these pages to discern the most essential aspects of my contribution to the mathematics of our time, via a comprehensive vision that chooses the forest over the trees - I've observed, not a victorious collection of "grand theorems", but rather a living spectrum of fertile ideas, which in their confluence have contributed to the same immense vision.(*)
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(*)That does not my work is lacking in major theorems, including those theorems which resolve questions posed by others, which no-one before me had known how to solve. (Some of these are reviewed in the note at the bottom of the page (***) pg. 554 or the note "The rising sea..." (R&S, #122) Yet, as I've already emphasized right at the beginning of this "promenade" (#6 "Vision and points of view), these theorems assume meaning for me only within the nourishing context of a grand theme initiated by one of those "fertile ideas". Their demonstration follows from them, as from a spring and effortlessly, even from their very nature, out of the "depths" of the theme that carries them - like the waves of a river appear to emerge calmly from the very depths of its waters, without effort or rupture. I've expressed the same idea, though with different images, in the footnote cited above, "The rising sea....".
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18. The Child and its Mother
When, in the course of writing this "preface", I began this promenade through my work as a mathematician, (with its brief sketches of "inheritors" (authentic), and "builders" (incorrigible)), a name suggested itself by which this incomplete preface could be suitably designated. Originally it was "The child and the builders". Over the course of several day s however, it became apparent that "the child” and "the builder" were one and the same person. This appellation thereby became, simply, "the child builder"; a name, indeed, not lacking in charm, with which I was well pleased. Yet it was revealed further along in the course of this reflection that this haughty "builder" or, (with more modesty), The child who plays at making houses was nothing more than one of the two avatars of the child- who-plays. There is, in addition, the child-who-loves-to-investigate-all -things, who delights in digging in and being buried by the sands, or in the muddy sludge, all those exotic, impossible surroundings... To indicate this change (if only for myself), I started to speak of him by means of the flashy word, the " pioneer"; followed by another more down to earth, though not lacking in prestige, the "explorer". I was then led to ask which, between the "builder" and the "pioneer-explorer", is the more masculine, the more enticing of the two? Heads or tails?
Following which, scrutinizing ever more closely, I beheld our intrepid "pioneer" who finds himself ultimately become a girl (whom I would have liked to dress up as a boy) - sister to pools, the rain, the fogs and the night, mute and virtually invisible from the necessity of staying always in the shadows – she whom one always forgets (when one is not inclined to mock her)... And I as well found opportunities as well, for days at a time, to forget her - to do so doubly, one might say: I tried to avoid seeing anything but the boy (he who plays at making homes)- and even when it became impossible all the same to deny the other, I still saw her somehow in the guise of a boy...
As a suitable name for my "promenade" in fact, it doesn't work at all. It's a phrase which is totally "yang", totally "macho", and it's lame. Not to appear biased it would have to also include the other But, strange as it may seem, the "other" really doesn't have a name. The closest surrogate would be "the explorer", but that too is a boy's name, and there's no hope for it. The language itself has been prostituted, it lays traps for us without our being aware of it, it goes hand in glove with our most ancient prejudices.
Perhaps one could make do with "the child-who-builds and the child-who-explores". Without stating that one is a "boy", the other a "girl", that it's a kind of single boy/girl who explores while building and while exploring builds.... Yet just yesterday, in addition to the double-sided yin-yang that both contemplates and explores, another aspect of the whole situation emerged.
The Universe, the World, let alone the Cosmos, are basically very strange and distant entities. They don't really concern us. It is not towards them that the deepest part of ourselves is drawn. What attracts us is an immediate and tangible Incarnation of them, that which is close, "physical", imbued with profound resonances and rich in mystery- that which is conflated with the origins of our being in the flesh, and of our species - and of That which at all times awaits us, silently and ever welcoming, "at the end of the road". It is She, the Mother, She who gives us birth as she gives birth to the World, She who subdues the urges or opens the floodgates of desire, carrying us to our encounter with Her, thrusting us forwards towards Her, to a ceaseless return and immersion in Her.
Thus, digressing from the road on this unanticipated "promenade", I found, quite by accident, a parable with which I was familiar, which I'd almost forgotten - the parable of The Child and the Mother. One might look upon it as a parable of "Life in Search of Itself". Or, at the simple level of personal existence, a parable of "Being, in its quests for things ".
It's a parable, and it's also the expression of an ancestral experience, deeply implanted in the psyche - the most powerful of the original symbols that give nourishment to the deepest levels of creativity. I believe I recognize in it, as expressed in the timeless language of archetypal images, the very breath of the creative power in man, animating flesh and spirit, from their most humble and most ephemerable manifestations to those which are most startling and indestructible.
This "breath", even like the carnal image that incarnates it, is the most unassuming of all things in existence. It is also that which is most fragile, the most neglected and the most despised...
And the history of the vicissitudes of this breath over the course of its existence is nothing other than your adventure, the "adventure of knowledge" in your life. The wordless parable that gives it expression is that of the child and the mother.
You are the child, issued from the Mother, sheltered in Her, nourished by her power. And the child rushes towards the Mother, the Ever-Close, the Well-Understood - towards the encounter with Her, the Unlimited, yet forever Unknowable and full of mystery...
This ends the "Promenade through the life's work of a mathematician"
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October 30, 2003
TRANSLATOR'S NOTE: This also ends my translation of Récoltes et Semailles until I can raise enough money to continue it. Since 1996 I have been essentially dependent upon welfare for survival. Just last month my cash stipend was reduced 43%, from $350 per month to a bare $200. I've observed that when the translation of R&S was once again put at the disposal of the public, the hits on my website went up by over 3,000 per month. At the same time not a penny has been received to help maintain the website, or assist with the production of the translation. Mathematicians are self-centered. I guess I don't have to say this; all I need to do is look into a mirror in the morning. But it's true. Small donations are as welcome as large ones. They will be immediately acknowledged by E-Mail. If you wish, your name will go onto a list of subscribers posted weekly at this location. Uploading the translation of R&S on the Ferment Magazine website will recommence once the benchmark of $5,000 (from a combination of donations or grants) has been received.
If anyone else is willing to pick up on the translation, I will be only too happy to either (i) provide a link to their website, or (ii) post their work in Ferment Magazine. A very sizable portion of the original French version of R&S is now available at Grothendieck Circle
Roy Lisker
8 Liberty Street#306 Middletown, CT 06457
risker@yahoo.com
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