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Récoltes et Semailles, Part I The life of a mathematician Reflections and Bearing Witness Alexander Grothendieck 1986 English Translation by Roy Lisker Begun December 13, 2002 By way of a Preface ..... written January 30th, 1986 Only the preface remains before sending Recoltes et Semailles to the printer. And I confess that I had every intention of writing something appropriate . So mething quite reasonable, for once. No more than 3 or 4 pages, yet well expres sed, as a way of opening this enormous tome of over a thousand pages. Somethin g that would hook the skeptical reader, that would make him willing to see wha t there was to be found in these thousand pages. Who knows, there may well be things here which might concern him personally! Yet that's not exactly my style, to hook people. But I was willing to make an exception in this case, just this once! It was essential if this book was to f ind an "editor crazy enough to undertake the venture" , ( of publishing this c learly unpublishable monster). Oh well, it's not my way. But I did my best. And not in a single afternoon, as I'd originally intended. Tomorrow it will be three weeks I've been at work on it, that I've watched the pages pile up. What has emerged can't in any sense be called a "preface" . Well, I've failed again. At my age I can't make myself over: I'm not made to be bought and sold! Even if I had every intention of be ing pleasing, to others and to myself ... What I've ended up with is a kind of long "promenade" with commentary, through my life's work as a mathematician. A promenade for the lay public - for those who "don't understand a thing about mathematics". And for myself as well, bec ause I've never embarked on such a stroll. I saw myself engaged in uncovering and in talking about things that have always been more or less proscribed. Coi ncidentally they are also those matters which I consider most basic to my work and my opus. These things have nothing to do with mathematical technicalities . You are the judge of my success in this enterprise, which I agree is really a bit insane. I will be satisfied if I have made you feel something of what I have felt, things which most of my colleagues don't know how to feel. It may b e that they have become to erudite, or received too many honors. Such things c ause one to lose contact with the essentials. In the course of this "Promenade through an opus" I will also be talking about my own life. And, here and there, of the purpose behind Recoltes et Semailles . Following the "Promenade" you will find a letter, ( dated May of the previou s year) . This letter was to be sent to my former students and my "old friends " in the world of mathematics. This also, is not technical in any sense. It sh ould be readable by anyone who has an interest in learning, via a living docum ent, about all of those odds and ends which have culminated in the production of Recoltes et Semailles. Even more than the Promenade, the Letter should give you an idea of a certain kind of atmosphere, that of the mathematical world i n its largest sense. And, also, (in the Promenade) , you may find my manner of expression a bit unusual, as you may find the mentality that naturally employ s such a style - one that is far from being understood by the rest of the worl d In the Promenade, and here and there in Récoltes et Semailles I will be speak ing of the nature of mathematical work. It is work that I understand very well from first hand experience. Most of what I say will apply equally well, I thi nk, to all creative labor, and all activities of discovery. It will apply at l east for what is known as 'intellectual' work, which is done mostly 'in one's head', and to writing. Work of this sort is distinguished by the hatching out and by the blossoming of our understanding of certain things which we are inte rrogating. To take an example in the other direction, passionate love is, also, driven by the quest for discovery. It provides us with a certain kind of understanding known as 'carnal' which also restores itself, blossoms forth and grows in dept h. These two impulses -that which animates the mathematician at his desk ( let 's say), and that which impels the lover towards the loved one - are much more closely linked than is commonly believed, or, let us say, people are inclined to want to believe. It is my wish that these pages of Récoltes et Semailles will make its reader aware of this connection, in his own work and in his dail y life. Most of the time In the course of this excursion we will be concerned with mat hematics itself, properly speaking. I will be saying almost nothing about the context in which this work takes place, or of the motivations of individuals w hich lie outside the work itself. This runs the risk of giving me, or the math ematician or scientist in general, a somewhat flattering image, and for that r eason distorted- the sort of thing one sees in speaking of the "grand passion" of the scientist, without restrictions. That is to say, something along the l ines of the grandiose "Myth of Science" ( with a capital S if you please!); th e heroic "myth of Prometheus" which writers have so often indulged in ( and co ntinue to do so) , for better or worse. Only the historians, and then not alwa ys, have been able to resist the seductions of this myth. The truth of the mat ter is that it is universally the case that, in the real motives of the scient ist, of which he himself is often unaware in his work, vanity and ambition wil l play as large a role as they do in all other professions. The forms that the se assume can be in turn subtle or grotesque, depending on the individual. Nor do I exempt myself. Anyone who reads this testimonial will have to agree with me . It is also the case that the most totally consuming ambition is powerless to m ake or to demonstrate the simplest mathematical discovery - even as it is powe rless ( for example) to "score" ( in the vulgar sense) . Whether one is male o r female, that which allows one to 'score' is not ambition, the desire to shin e, to exhibit one's prowess, sexual in this case. Quite the contrary! What brings success in this case is the acute perception of the presence of so mething strong, very real and at the same time very delicate. Perhaps one can call it "beauty", in its thousand-fold aspects. That someone is ambitious does n't mean that one cannot also feel the presence of beauty in them; but it is n ot the attribute of ambition which evokes this feeling.... The first man to discover and master fire was just like you and me. He was nei ther a hero nor a demi-god. Once again like you and me he had experienced the sting of anguish, and applied the poultice of vanity to anaesthetize that stin g. But, at the moment at which he first "knew" fire he had neither fear nor va nity. That is the truth at the heart of all heroic myth. The myth itself becom es insipid, nothing but a drug, when it is used to conceal the true nature of things. I intend in Récoltes et Semailles to speak of both aspects : of the passion f or knowledge, and the passion of fear and the antidotes of vanity used to curb it. I make the claim that I understand , or at least am well acquainted with, the passion for knowledge. ( Yet perhaps one day I will discover, to my amaze ment, to what extent I've been deceiving myself) . Yet when it comes to fear a nd vanity, and the insidious ways in which these block creativity, I am well a ware that I have not gotten to the root of this enigma. Nor do I know if I wil l ever see through to the end of this myself in the years remaining to me. Over the course of writing Récoltes et Semailles there emerged two images, re presenting two fundamental aspects of the human adventure: These are the child ( alias the worker ), and the boss . In the Promenade on which we are about t o embark, we will be dealing almost exclusively with the child. He also figure s in the section entitled "The child and the Mother". The meaning of this term will, I trust, become clear as we proceed. Yet in the remainder of the work it is the boss who will be at center-stage: h e isn't the boss for nothing! To be more precise, one isn't talking about a si ngle boss, but of various bosses of different enterprises being maintained con currently. At the same time, these bosses have a way of resembling one another in their essential nature. Once one begins to talk about bosses, there have to be villains. In Part I of the section entitled "Complacency and Restoration" (Fatuité et Renouvellement ), which comes right after the introductory material ( Prelude in 4 Movements) , it is I, above all, who am the "villain"! In the remaining 3 sections, its t he others. Everyone gets a turn! In other words one can expect to find, along with a number of more or less pro found philosophical reflections and some 'confessions'(without contrition), se veral "acid sketches" (portraits au vitriol- to use the expression of one of m y colleagues who has found himself somewhat mistreated), as well as a host of vigorous "operations" which have not been sanitized. Robert Jaulin* has assure d me ( only partly joking) that what I'm doing in Récoltes et Semailles is a kind of 'ethnography of the mathematics community' ( or perhaps the sociology) ------------------------------------------------------------------------------ -- (*) Robert Jaulin is an old friend of mine. It is my understanding that he, vi s-a-vis the community of ethnologists, is regarded, ( as I am in the mathemati cs establishment) as something of a 'black sheep'. ------------------------------------------------------------------------------ -- It is extremely flattering of course to learn, ( without having been aware of it), that one is engaged in real scholarship! It is a fact that in the inquiry portion of this enterprise I have watched (at some risk to myself) the passag e across these pages of a considerable portion of the mathematics establishmen t, as well as friends and colleagues of less exalted status.That has come back to me over the last few months during which I've started sending out a provis ional first edition of Récoltes et Semailles. Unquestionably my testimony has had the effect of tossing a brick through a gl ass window! Echoes of every sort ( save that of boredom) have resonated from e verywhere. Frankly this was not what I'd expected. And there's been lots of si lence too, the kind that speaks volumes Clearly I still have a lot to learn, about all the things going on in the priv ate retreats of others, such as ex-students and those former colleagues who se em to be doing pretty well for themselves ( my apologies ?I meant to say in th e 'sociology of the mathematics community'!) To all those who, in their own wa y, have contributed to this 'sociological research' with which I occupy my eld erly days, I of course express my profoundest gratitude. Needless to say I have been most receptive to the enthusiastic responses. Ther e have also been those colleagues , rare enough, who have shared with me their feelings and their experiences about the state of crisis, and the extreme deg radation, which lies at the heart of the contemporary mathematics community, o f which they are members. Among those who, outside of this circle, have been among the first to give thi s testimony a warm reception I wish to single out Sylvie and Catherine Chevall ey(*), Robert Jaulin, Stéphane Deligeorge, Christian Bourgeois. ------------------------------------------------------------------------------ -- (*)Sylvie and Catherine Chevalley are the widow and daughter of Claude Chevall ey, the colleague and friend to whom I've dedicated the central core of Récol tes et Semailles ( R&SIII,"The Key to the Yin and the Yang"). I will be speaki ng of him in many places, and of his role in my personal journey. ------------------------------------------------------------------------------ -- Well, here we go! Let us begin our Promenade through the work of a lifetime,a s a way also of speaking of that life itself. It will be a long voyage indeed, of more than a thousand pages, each of them filled with substance. It has req uired the whole of my life to make this voyage, which is far from finished, an d more than a year to reconstruct it, page upon page. The words have sometimes come hesitantly, as they attempt to give the full import of my experience, th e understanding of which has also come hesitantly ?like the ripe grape buried in the winepress that may offer resistence to the force applied to crush it.Ye t even at those times when it appeared that words were pouring out, tumbling o ver one another in their urge for release, they were not being strewn at rando m on the page. Each of them has been carefully weighed, either at the moment o f their emergence or in subsequent consideration, and appropriately modified i f too light or too heavy. Thus, don't expect that this reflection-witnessing-voyage will make for facile reading, in a day or even in a month. It is not intended for the reader who w ishes to come to the end of it as quickly as possible. One can't really speak of "endings", much less "conclusions" in a work like Récoltes et Semailles , no more than one finds such things in my life or in yours. Think of it like a wine fermented in the depths of someone's being for a lifet ime. The last glass will be neither better nore worse than the first, or the h undredth. They are all alike, and they are all completely different. And if th e first goblet is spoiled, the whole vat from which it comes is likewise spoil ed. Far better to drink good water than bad wine! Yet, when one finds a good wine, it is best to sip it slowly, and not when one is one the run. ------------------------------------------------------------------------------ -- Introduction ------------------------------------------------------------------------------ -- Return to Grothendieck Folder Introduction It must have been in July of 1984 that I had an extraordinary dream. When I us e the expression "extraordinary", I refer only to the impression that it made on me afterwards. The dream itself appeared as the most natural thing in the w orld, without any sort of fanfare ?to such an extent that, even after awakenin g I attached no importance to it, and buried it somewhere in the secret dungeo n of the unconscious so as to get on to the business at hand. Since the previous day I'd been reflecting on my relationship to mathematics, indeed it was probably the first time ever that I'd thought to consider this m atter seriously. Indeed if I was doing so now, it was only because I was force d into it. So many strange, even violent things had happened over recent month s and years, one might call them veritable explosions of the passion for mathe matics that continue to erupt without showing any sign of diminishing, that I simply could not proceed further without taking an overview of what had been g oing on. The dream I'm referring to had no scenario, no specific acts or activities. It contained but a single frozen image, one that was at the same time remarkably alive. It was a human head seen in profile, scanned from left to right. The h ead was of a mature man, beardless, with wild head wrapped around its brow lik e a bright powerful halo. The strongest impression made by this head was of a joyous, youthful vitality, which seemed to spring directly from the supple and vigorous arching of its neck (sensed more than seen). The facial expression w as more that of a mischievous delinquent than of a responsible or settled adul t, thrilled by the recollection of some trick he'd gotten away with or was abo ut to do. It gave off an intense love of life, playful, content with itself. Nobody else was present, no-one to play the role of observer, and "I" to look at or contemplate this being, of whom one saw only the head. Yet the perceptio n of this head, or let us say of the atmosphere which it evoked, was extremely intense. Nor was anyone else present to record impressions, comment on them, or give a name to the person being observed, to call him "this or that". There was only this intensely vital object, the man's head, and an awareness of tha t vitality. When I awoke and reviewed the various dreams which had passed that night, the one with the man's head did not seem of any particular significance, there was nothing in it that might make me cry out: you ought to be looking at me! Revi ewing this dream in the quiet comfort of my bed, I was driven by the natural d esire to put a name to this apparition. Nor did I have far to search: once the question was posed it was more than obvious that the head I'd seen in my drea m was none other than my own. It's not a bad thing, I told myself, it takes some doing to see one's own head in a dream as if it were that of another! The dream gave the impression of ha ving arisen by accident, much as when one finds a 4-leaf clover, (or even one of 5 leaves), yet aroused no other reaction, so that shortly afterwards I felt free to go on my way as if nothing had happened. That's more or less the way it happened. Happily, as often occurs in situation s of this sort, as a way of putting my conscience to rest I wrote it down, whi ch was enough to set off an meditation that would hook up with that which I'd begun the previous evening. Then, little by little, the thoughts of that day b egan to involve me more and more with the event of that dream, its single cohe rent image, and the message it carried for myself. This is not the proper place to expand on all that this day's meditation taugh t and delivered ?to be more precise, that which the dream taught and delivered once I'd put myself in a receptive state of mind which enable me to gather wh at it was trying to tell me. I can say however that the first fruit of that dr eam, and of this state of receptivity, was a spontaneous upwelling of fresh en ergy. It was this energy which would give me over the following months, the st amina to persist in the long meditation that, by means of patient and persiste nt work, resulted in overcoming many internal resistances and opinions. Its been five years since I began to pay attention to a certain number of my d reams. This was the first "messenger dream" which didn't have the usual charac teristics of such a dream, which generally combines impressive scenery and an heightened intensity of vision which can at times be overwhelming. This was fa r more subtle, low key, with nothing in it to force one to focus one's attenti on, the very model of discretion- one could, it appeared, take it or leave it. A few weeks later I received a messenger dream of the conventional sort, runn ing the gamut of drama, even of savagery, which had the effect of bringing to a sudden end a long period of frantic mathematical activity. The only connecti on between the two dreams was that neither in the one nor in the other was the re any kind of observer. Like the parabolic trajectory of a stone under the ef fect of gravity, this dream was showing me something that was going on in my l ife, quite apart from any influence or even knowledge of it on my part ?indeed , things which I'd gone out of my way to deny. It was this dream, above all, w hich impressed upon me the urgency for a labor of internal reflection. Beginni ng a few weeks later, this labor continued for another six months. I will have occasion to speak of it in the latter part of the section entitled "Recoltes et Semailles" which opens the present volume and gives its name to the entire opus. (*) ------------