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 matter seriously. Indeed if I was doing so now, it was only because I was forced into it. So many strange, even violent things had happened over recent months and years, one might call them veritable explosions of the passion for mathematics 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 going 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 head was of a mature man, beardless, with wild head wrapped around its brow like 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 was more that of a mischievous delinquent than of a responsible or settled adult, thrilled by the recollection of some trick he'd gotten away with or was about 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 perception 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 that 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! Reviewing this dream in the quiet comfort of my bed, I was driven by the natural desire 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 dream 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 having 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 situations of this sort, as a way of putting my conscience to rest I wrote it down, which 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 began to involve me more and more with the event of that dream, its single coherent image, and the message it carried for myself.
This is not the proper place to expand on all that this day's meditation taught and delivered Ð 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 what it was trying to tell me. I can say however that the first fruit of that dream, and of this state of receptivity, was a spontaneous upwelling of fresh energy. It was this energy which would give me over the following months, the stamina to persist in the long meditation that, by means of patient and persistent 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 dreams. This was the first "messenger dream" which didn't have the usual characteristics of such a dream, which generally combines impressive scenery and an heightened intensity of vision which can at times be overwhelming. This was far more subtle, low key, with nothing in it to force one to focus one's attention, 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, running 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 connection between the two dreams was that neither in the one nor in the other was there any kind of observer. Like the parabolic trajectory of a stone under the effect of gravity, this dream was showing me something that was going on in my life, quite apart from any influence or even knowledge of it on my part Ð indeed, things which I'd gone out of my way to deny. It was this dream, above all, which impressed upon me the urgency for a labor of internal reflection. Beginning 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. (*)
At the same time the long meditation on the "past of the life of a mathematician" has been drawing to its close. In retrospect, speaking honestly, the 3 years that have gone by since the appearance of this dream have been years of crystallization and maturation as I've moved towards the enunciation of a message that is both simple and lucid.
That dream showed me to myself, "as I am" . Equally, it is clear that the being in my waking life is not at all the same as the one that the dream revealed to me. Obstacles and restrictions going back a long distance have obstructed ( and continue to obstruct) the possibility for me to be simply myself. Over these years, even though the recollection of this dream came to me only rarely, it must certainly have acted on me in several ways. One should not think of it as a kind of model or ideal which I felt myself obliged to imitate, but merely of a discrete reminder of a kind of joyous simplicity which at one point "was me", which reveals itself under numerous guises, which seeks to be liberated from that which weighs upon it, and which continues to blossom forth. This dream was the tie, both delicate and vigorous, between a present still laden down with past burdens, and an immanent "tommorow" which takes the form of a seed, a "tommorow" which is in my present, and which has certainly always been within me. Certainly if in those weeks this rarely recollected dream had been present within me , it had to be the level of thinking that is weighed and analyzed. I should have realized that the work that I was engaged on and bringing to a conclusion, constituted a new step in the direction of the message about myself which it conveyed.
This is, at the present moment, my sense of the meaning of Récoltes et Semailles, and of the intensive work that had occupied the last two months. Only now, after it has been finished, have I fully realized the importance of what I was doing. In the course of this effort I've known many moments of happiness, a happiness that has often been mischievous, comic and exuberant. And there have also been moments of sadness, those moments when I had to relive all the frustrations and sorrows which have been my lot over recent years. But I have not known a single moment of bitterness. I end this work with the total satisfaction of one who has brought a difficult work to completion. I have evaded nothing, not matter how minor, which in my heart I knew had to be said, so I am not left with the dissatisfaction of having left something undone.
It has always been clear to me while I was writing this testimonial that it could not please everybody. In fact, it's quite possible that I've found a way to disappoint everyone. However my intention has only been to cast an overview on the simple details, the basic events of my past ( and present also) as a mathematician, to convince myself ( better late than never), without reservation or the shadow of a doubt, of what they were and what they are; and, once having taken this road, to describe what I've learned as simply as possible.
When at the end of several weeks I'd saw that it was approaching the first 100 pages, that introduction was promoted to the status of "introductory chapter"! Another two weeks went by, by which time the dimensions of this "chapter" had grown beyond those of all other chapters in the work being prepared ( all, with the exception of the final one , complete at the time of writing these lines) , I finally understood that its proper place was not in a book about mathematics, in which this testimonial would be quite inappropriate. It deserved a volume all by itself, which I intended to be Volume I of those "Reflections on Mathematics" which I intend to develop in the years to come, as sequel to "Defining Stacks ".
I do not want to claim that Récoltes et Semailles, ( which is the first volume in the series of "Mathematical Reflections" ( and which will be followed by two or three volumes of "Defining Stacks" , to begin with), )ought to be considered as a kind of "introduction" to the Reflections . I rather see this first volume as the foundation of all that is to come, or , to express it better, something to set the tone, the spirit in which I intend to undertake this new voyage, that which I intend to pursue over the years to come, which will lead me I know not where.
To round out these comments about the subject of the major part of the present volume, I will give several indications of a practical nature. The reader should not be surprised to find in the text of Récoltes et Semailles that there are references to the "present volume" - by which is meant the first volume ( History of Models) of "Defining Stacks", of which I am still in the process of writing the introduction. I have not bothered to "correct" these passages because I don't want to lose the spontaneity with which the text was written, nor the authenticity of its evidence, not only on a distant past, but of the moment at which it is being written.
It is for this same reason that my revisions of the first draft of the text have been restricted to obvious clumsiness in the style, or of a way of stating things that might be so confusing as to seriously compromise one's understanding of what is being stated. These revisions have sometimes led me to a more subtle or clearer grasp of the matter. Modifications of greater weight, to nuance, render precise, complete or sometimes to correct what is stated in the text, have been put into over fifty numbered footnotes, grouped together at the end of this treatise, and which constitute more than 25% of the text. (*)
As one might expect, to some of these very long footnotes have been added several sub-notes. These are then included at the end of the text of the original footnote, with the same sort of cross-reference, save for very short notes which can be found on the same page, with cross-references such as (*) or (**) .
It was a source of delight for me to adjoin names to each section of this text, as well as to some of the more substantial notes. Indeed it was indispensable that I do so in order for me to resituate myself in the text. It goes without saying that these names were invented afterwards, given that when I began a section or a footnote I had no idea where my thoughts would lead. It was the same de fortiorti with respect to names ( such as "Work and Discovery", etc. ) by which are designated the 8 subdivisions , I through VIII, in which, as a afterthought, the fifty sections of text were organized.
With regards to the content of these 8 subdivisions I will limit myself to a few very brief observations. The first two parts , I ( Work and Discovery) , and II ( The Dream and the Dreamer) , contain the elements of a meditation on the nature of mathematical work, and on the nature of the adventure of discovery in general. My own person is engaged in a fashion far more haphazard and indirect than in the parts that follow. These are the ones which, above all, have the quality of documentation and of meditation.
Parts III to VI are above all are a backward look and a reflection on my past as a mathematician , "in the world of mathematics", between 1948 to 1970. The motivation inspiring this meditation has been the desire to understand that past, as part of my effort to understand and assimilate a present of which certain elements may appear deceptive or perplexing.
Parts VII ( The child at play ), and VIII ( The solitary adventure) are mostly concerned with the evolution of my relationship to mathematics between 1970 up to the present, that is to say, from the time I left the "world of mathematics" for good. Herein I've examined the motives, forces and circumstances that have led me ( to my great surprise) to once again pick up a public role in mathematics ( by writing and publishing Mathematical Reflections), after an interruption of over 13 years.
The "Thematic Outline" ("Esquisse thematique des principaux travaux mathematiques de A. Grothendieck")was written in 1972 in defense of another candidacy, ( for a professorship with the College de France) . It outlines, in a thematic catalogue, those things which I believe to be my principal contributions to mathematics. This text reflects my tendencies at the time it was written, at a time when my interest in mathematics was extremely marginal, to say the least. Thus this outline is little more than a dry and methodical enumeration ( which, happily, made no attempt to be exhaustive.)
It is completely lacking in vision or passion, as if the things I was talking about, ( through some notion of acquitting my conscience ( which indeed describes what I tended to feel)) had never been animated by a living vision, nor by any passion to bring them into the light of day when they were still behind their curtains and shadow and fog.
Despite this I've decided to include this demoralized document because, I fear, to silence ( imagining that it would be possible to do so) certain highly placed colleagues of a certain sort who, since my departure from the world of mathematics, choose to be somewhat disdainful of what they've amiably dismissed as my grothendieckeries .
Apparently this has become a synonym for wasting precious time on things beneath the concern of a serious and mature mathematician. Possibly this "indigestible digest" will impress them as something to be taken more seriously! As for the works I've produced under the force of passion and vision, it is only to be expected that those persons whom the world supports and acclaims, are insensitive to the things which have enchanted me. If I have written for others besides myself, it has been for those who have not stingily guarded their time and their persons, who have steadfastly persevered in examining those self-evident things which others dare not look at, who rejoice in the inimitable beauty of each of those things I've discovered, things distinguished for their intrinsic beauty from whatever came before.
If I were to attempt to place the three texts of the present volume in relation to one another, ( and the role of each in the voyage first undertaken with the Mathematical Reflections), I would say that the reflection/testimonial "Ré et Semailles" , reflects and describes the spirit in which I undertook this voyage, and which gave it meaning. The "Outline of a Programme" details my sources of inspiration, which set a direction, if not exactly a destination, for this voyage into the unknown, a bit like the needle of a compass, or a vigorous Ariadne's Thread. Finally the "Thematic Outline" is a rapid run-through of a certain amount of old baggage, acquired in my past as a mathematician up to 1970, a part of which at least may be serviceable to this or that stage of my voyage ( such as for example my reflections on Cohomological and Topos Algebras, which are indispensable for me in the conception of "Defining Stacks" ).
The order of these 3 texts, as well as their respective lengths, are an accurate reflection ( with any deliberate intention on my part) of the relative importance I've assigned to them in the journey, of which the first stage is now approaching its end.
For a certainty I am unable to predict how this voyage will turn out, for it is something which is to be discovered in the course of my pursuing it. Right now I have no grand itinerary, and I don't think there will be one. As was stated before, the major themes which I'm pretty certain will serve as the inspiration for this reflection are essentially sketched in the "Outline for a Programme", which is the 'orienting text' ("texte-boussole). Among these themes is included the major theme of "Defining Stacks" that is to say the "stack" itself, which I hope to cover in the next year, in two or perhaps three volumes. With respect to this theme I wrote the following in the "Outline": "...it is something like a debt that I have contracted against myself vis-a-vis my scientific past, over a 15-year period ( between 1955 and 1970). The constant refrain of this period had been the laying of the foundations of Algebraic Geometry through the development of the tools of co-homology.
It is this, in the list of all the themes I expect to touch on, which has the strongest connection with my scientific past. And it is this which I've most regretted over the last 15 years, as the most flagrant omission among all those which I abandoned by leaving the mathematical scene, which none of my students or former friends have taken the trouble to correct. For more details on this work in progress, the interested reader may turn to the relevant section in the "Outline", or to the introduction ( the 'real' one , this time!) of the first volume of "Defining Stacks".
Another legacy of my scientific past that is particularly close to me, is of course the concept of the "motive", waiting its turn to come out of the night into which it has been enshrouded since it first made its appearance over a good 15 years ago. It is not impossible that I may try to complete my work on the foundations of this subject, if no-one in a better position than I am, ( younger perhaps, with fresh tools and knowledge) sees it worth his while to undertake such a project in the coming years.
I would like to use this occasion to point out that the fate of the notion of the "motive", and of several others which I brought into the light of day, those that are the most powerful and fertile, will be the subject for a discussion of about 20 pages, forming the longest footnote ( and one of the last) in Récoltes et Semailles (*)
At one point I'd thought of separating this gigantic footnote from the text from which it is derived, and placing it in the "Thematic Outline". This would have had the further advantage of putting it in proper perspective, and to inject a bit of life into a text that may appear to be something of a dry catalogue. I decided not to do this out of a wish to maintain the integrity of a testimonial for which this "mega-footnote", whether or not it pleases me, forms an integral part.
With regard to what is stated in Récoltes et Semailles regarding my intentions when I began the "Mathematical Reflections", I need only add one more thing: It suffices to quote what has already been written in a footnote ( The snobbery of youth - or the defenders of purity ) : "Through my whole life my ambition as a mathematician, or rather my passion and joy, has ever been to uncover self-evident truths. This is also my sole ambition in this present work.(Defining Stacks) . It remains my unique ambition for this new voyage which, a year later, is being pursued in the "Mathematical Reflections". And it is this same in these Récoltes et Semailles which ( for my readers at least, if there are any) open this voyage. I would like to conclude this introduction with a few words on the pair of dedications made at the beginning of Récoltes et Semailles. The dedication " to those former students of mine to whom I've given the best of myself- and also the worst", had been decided upon since at least the previous summer, particularly when I was writing the first four sections of what was still envisaged only as an introduction to a work in mathematics. I knew very well, in other words, and in fact had known it for several years already, that that "worst part of myself" would have to be looked into - and that it was now or never! (Although I'm certain that this 'worst part' would led to another opus of a minimum of 200 pages.)
On the other hand, the dedication "to my elders", only came to me in the course of the writing, just like the name of this whole reflection (which has also become that of one of its volumes). I came to realize the great importance of their role in my life as a mathematician, a role whose effects are still very apparent today. This will no doubt become clear in the pages that follow, so there is no need to elaborate on this subject. These "elders" are, in the approximate order in which they entered my life from the age of 20 onwards: Henri Cartan, Claude Chevalley, André Weil, Jean-Pierre Serre, Laurent Schwartz, Jean Dieudonné , Roger Godement, and Jean Delsarte. Those persons encountering me for the first time may not be aware of the degree of enthusiasm and good-will with which all of them welcomed me, or the extent to which so many of them have continued in the solid friendship and affection.
Here is the place also to mention Jean Leray, whose warm reception at the time of my first encounters with the world of mathematics (1948/49) was also an important source of encouragement. This reflection has made me realize my debt of gratitude towards each of these men, " from another world and with another destiny". This debt is in no sense a burden. To uncover it has been a source of joy, and lightened my spirit. (The end of March, 1984)
. By the force of circumstances and through the inner logic of this retrospective, I have been led to have to implicate others besides myself. The person in question is, more than all others myself excepted, a man bound to me with ties of affection for close to 20 years. Of him I have written (in the form of a euphemism(*) , that he had "taken the part of a student" in the early years of this close friendship based on a common passion.
I do not apologize for making public, in these writings, a personal reflection on a personal relationship, nor to implicate someone without consulting him beforehand.It seems important that me that a situation which, for all too long, has been shrouded in confusion and obscurity, be brought into the light of day for examination. This being said, I am writing a testimonial, one that is certainly subjective and which does not claim to exhaustively treat of so delicate a situation, nor to be free of errors. Its principal merit, ( like that of my earlier publications, or those on which I'm working presently), is that it exists to be consulted by those persons to whom it will be of interest. My concern is neither to convince, nor to shield myself from the possibility of being mistaken in documentation only those things which one can assume are taken for granted. My concern is only to be faithful to the truth as I see it, in order to get to the bottom of certain things and render them more understandable.