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 . Something quite reasonable, for once. No more than 3 or 4 pages, yet well expressed, as a way of opening this enormous tome of over a thousand pages. Something that would hook the skeptical reader, that would make him willing to see what 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 find an "editor crazy enough to undertake the venture" , ( of publishing this clearly 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 being 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, because 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. Coincidentally 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 be that they have become to erudite, or received too many honors. Such things cause 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 previous 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 should be readable by anyone who has an interest in learning, via a living document, 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 in 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 employs such a style - one that is far from being understood by the rest of the world

In the Promenade, and here and there in Récoltes et Semailles I will be speaking 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 think, to all creative labor, and all activities of discovery. It will apply at least 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 interrogating.

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 depth. 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 daily life.

Most of the time In the course of this excursion we will be concerned with mathematics itself, properly speaking. I will be saying almost nothing about the context in which this work takes place, or of the motivations of individuals which lie outside the work itself. This runs the risk of giving me, or the mathematician or scientist in general, a somewhat flattering image, and for that reason 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 lines of the grandiose "Myth of Science" ( with a capital S if you please!); the heroic "myth of Prometheus" which writers have so often indulged in ( and continue to do so) , for better or worse. Only the historians, and then not always, have been able to resist the seductions of this myth. The truth of the matter is that it is universally the case that, in the real motives of the scientist, of which he himself is often unaware in his work, vanity and ambition will play as large a role as they do in all other professions. The forms that these 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 make or to demonstrate the simplest mathematical discovery - even as it is powerless ( for example) to "score" ( in the vulgar sense) . Whether one is male or female, that which allows one to 'score' is not ambition, the desire to shine, 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 something 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 doesn't mean that one cannot also feel the presence of beauty in them; but it is not the attribute of ambition which evokes this feeling....

The first man to discover and master fire was just like you and me. He was neither 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 sting. But, at the moment at which he first "knew" fire he had neither fear nor vanity. That is the truth at the heart of all heroic myth. The myth itself becomes 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 for 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 amazement, to what extent I've been deceiving myself) . Yet when it comes to fear and vanity, and the insidious ways in which these block creativity, I am well aware that I have not gotten to the root of this enigma. Nor do I know if I will 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, representing 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 to embark, we will be dealing almost exclusively with the child. He also figures 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: he isn't the boss for nothing! To be more precise, one isn't talking about a single boss, but of various bosses of different enterprises being maintained concurrently. 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 the others. Everyone gets a turn!

In other words one can expect to find, along with a number of more or less profound philosophical reflections and some 'confessions'(without contrition), several "acid sketches" (portraits au vitriol- to use the expression of one of my colleagues who has found himself somewhat mistreated), as well as a host of vigorous "operations" which have not been sanitized. Robert Jaulin* has assured 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, vis-a-vis the community of ethnologists, is regarded, ( as I am in the mathematics 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 passage across these pages of a considerable portion of the mathematics establishment, 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 provisional first edition of Récoltes et Semailles.

Unquestionably my testimony has had the effect of tossing a brick through a glass window! Echoes of every sort ( save that of boredom) have resonated from everywhere. Frankly this was not what I'd expected. And there's been lots of silence too, the kind that speaks volumes

Clearly I still have a lot to learn, about all the things going on in the private retreats of others, such as ex-students and those former colleagues who seem to be doing pretty well for themselves ( my apologies I meant to say in the 'sociology of the mathematics community'!) To all those who, in their own way, have contributed to this 'sociological research' with which I occupy my elderly days, I of course express my profoundest gratitude.

Needless to say I have been most receptive to the enthusiastic responses. There 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 degradation, which lies at the heart of the contemporary mathematics community, of which they are members.

Among those who, outside of this circle, have been among the first to give this testimony a warm reception I wish to single out Sylvie and Catherine Chevalley(*), Robert Jaulin, Stéphane Deligeorge, Christian Bourgeois.

(*)Sylvie and Catherine Chevalley are the widow and daughter of Claude Chevalley, the colleague and friend to whom I've dedicated the central core of Récoltes et Semailles ( R&SIII,"The Key to the Yin and the Yang"). I will be speaking 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,as 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 required the whole of my life to make this voyage, which is far from finished, and 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, the 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.Yet even at those times when it appeared that words were pouring out, tumbling over one another in their urge for release, they were not being strewn at random on the page. Each of them has been carefully weighed, either at the moment of their emergence or in subsequent consideration, and appropriately modified if 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 wishes 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 lifetime. The last glass will be neither better nore worse than the first, or the hundredth. They are all alike, and they are all completely different. And if the first goblet is spoiled, the whole vat from which it comes is likewise spoiled. 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.

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