<收获与播种> 第一部分 英文机翻 An English version of the first part of *Recoltes et Semailles*, translation done by AI

致潜在读者：

​ 这儿有部分内容的英文翻译，别人已经做过的工作我不再做。

​ 我把文本粗看了一遍，作了些修改并填上了粗体和斜体字，并且作了些注释，标上 translator's note（更准确些说应该是校对者注——but whatever.）。不过可能还有些错误，包括很愚蠢的错误。

​ 以下是正文。

FIRST PART. FATUITY AND RENEWAL

To those who were my elders, who welcomed me fraternally into the world that was theirs and that became mine

To those who were my students, to whom I gave the best of myself - and also the worst...

I WORK AND DISCOVERY

Contents

  1. The child and the good lord
  2. Error and discovery
  3. Unspeakable labors
  4. Infallibility(of others) and contempt(of oneself)

June 1983

​ (1) The child and the good lord

[◊ 1] The mathematical notes on which I am now working are the first for thirteen years that I have intended for publication. The reader will not be surprised that, after a long silence, my style of expression has changed. This change of expression is not, however, the sign of a change in style or working method (1), and even less that of a transformation that has taken place in the very nature of my mathematical work. Not only has it remained the same, but I have become convinced that the nature of the work of discovery is the same from one discoverer to the next, that it transcends the differences created by infinitely varying conditioning and temperament.

Discovery is a child's privilege. It's the little child I'm talking about, the child who is not yet afraid of making mistakes, of looking stupid, of not being serious, of not being like everyone else. Nor is he afraid that the things he looks at have the bad taste of being different from what he expects them to be, from what they should be, or rather: from what it is well understood that they are. He ignores the mute and unwavering consensus that is part of the air we breathe - that of all sensible people and well known as such. God knows there have been many sensible people known as such since the dawn of time!

Our minds are saturated with a heterogeneous ‘knowledge’, a tangle of fears and laziness, cravings and prohibitions; of information for all purposes and push-button explanations - a closed space where information, cravings and fears pile up without the wind of the open sea ever blowing in. With the exception of routine know-how, it would seem that the main role of this ‘knowledge’ is to evacuate a living perception, a grasp of the things of this world. Its effect is above all one of immense inertia, of an often crushing weight.

The little child discovers the world as it breathes - the ebb and flow of its breath makes it welcome the world into its delicate being, and projects itself into the world that welcomes it. Adults also discover, in those rare moments when they have forgotten their fears and their knowledge, when they look at things or themselves with eyes that are wide open, eager to know, with new eyes - the eyes of a child.

[◊ 2] God created the world as he discovered it, or rather he creates the world eternally, as he discovers it - and he discovers it as he creates it. He created the world and creates it day after day, repeating himself millions and millions of times, without respite, groping his way, making millions and millions of mistakes and rectifying his aim, without tiring... Each time, in this game of probing into things, of the response of things (‘that's not a bad shot’, or : Every time, in this game of probing into things, the response from things (‘that's not a bad idea’, or ‘you're really messing up’, or ‘it's working like clockwork, keep it up’), and the new probe rectifying or repeating the previous probe, in response to the previous response..., at every turn in this infinite dialogue between the Creator and Things, which takes place at every moment and in every place of Creation, God learns, discovers, gets to know things more and more intimately, as they take on life and form and are transformed in His hands.

Such is the process of discovery and creation, such seems to have been from all eternity (as far as we can tell). It has been like this, without man having to make his late entrance on the scene, barely a million or two years ago, and get his hands dirty - with, in recent times, the unfortunate consequences we know about.

Sometimes one of us discovers one thing or another. Sometimes we rediscover in our own lives, with wonder, what it is to discover. Everyone has everything they need to discover everything that attracts them to this vast world, including this marvellous capacity within them - the simplest, most obvious thing in the world! (Yet it's something that many of us have forgotten, just as we have forgotten to sing, or to breathe as a child breathes...)

Everyone can rediscover what it is to discover and create, and no one can invent it. They were there before us, and they are what they are.

​ (2) Error and discovery

To come back to the style of my mathematical work itself, or its ‘nature’ or ‘approach’, it is now like those that the good Lord himself taught us without words, God knows when, long before we were born perhaps. I do as he did. It's also what everyone does instinctively, as soon as curiosity pushes them to know something of all things, something invested from then on by this desire, this thirst...

[◊ 3] When I'm curious about something, mathematical or otherwise, I question it. I question it, without worrying whether my question is perhaps stupid or whether it will appear so, without it being at all costs carefully weighed. Often the question takes the form of an assertion - an assertion which, in truth, is a sounding board. I believe my assertion more or less, depending of course on where I am in my understanding of the things I'm looking at. Often, especially at the beginning of a research project, the assertion is downright false - but you had to make it to be convinced. Often, all you had to do was write it down and it was obvious that it was false, whereas before writing it down there was a blur, like a feeling of unease, instead of this obviousness. Now you can come back to it with less ignorance, with a question that's perhaps a little less ‘off the mark’. Even more often, the statement taken literally turns out to be wrong, but the intuition that, clumsily still, tried to express itself through it is right, although still vague. Little by little, this intuition will be decanted from an equally shapeless gangue of false or inadequate ideas, it will gradually emerge from the limbo of the misunderstood that only asks to be understood, of the unknown that only asks to be known, to take on a form of its own, refining and sharpening its contours, as the questions I ask of these things in front of me become more precise or more relevant, to pin them down more and more closely.

But it can also happen that through this process, repeated probing converges on a certain image of the situation, emerging from the mists with sufficiently marked features to lead to the beginnings of a conviction that this image does indeed express reality - when, however, this is not the case, when this image is tainted by a major error, likely to distort it profoundly. The work, sometimes laborious, which leads to the detection of such a false idea, starting from the first ‘take-offs’ observed between the image obtained and certain obvious facts, or between this image and others which also had our confidence - this work is often marked by a growing tension, This work is often marked by a growing tension as we approach the crux of the contradiction, which at first seems vague but then becomes increasingly blatant - until finally it explodes, with the discovery of the error and the collapse of a certain vision of things, which comes as an immense relief, like a liberation. The discovery of error is one of the crucial [◊ 4] moments , a creative moment of all, in any work of discovery, whether mathematical or self-discovery. It is a moment when our knowledge of the thing suddenly probed is renewed.

**Fearing error and fearing truth are one and the same thing. **He who fears error is powerless to discover. It is when we fear being wrong that the error in us becomes immovable as a rock. Because in our fear, we cling to what we once thought was ‘true’, or to what has always been presented to us as true. When we are driven, not by the fear of seeing an illusory security vanish, but by a thirst for knowledge, then error, like suffering or sadness, passes through us without ever becoming fixed, and the trace of its passage is renewed knowledge.

​ (3) Unspeakable labors

It is surely no coincidence that the spontaneous process of all genuine research never appears in the texts or discourse that are supposed to communicate and transmit the substance of what has been ‘found’. More often than not, texts and speeches are confined to recording ‘results’, in a form that must appear to ordinary mortals as so many austere and unchanging laws, inscribed from all eternity on the granite tables of some sort of giant library, and dictated by some omniscient God to the initiates-scribes-wise men and women; to those who write learned books and no less learned articles, those who pass on knowledge from the pulpit, or in the more restricted circle of a seminar. Is there a single textbook, a single manual for use by schoolchildren, secondary school pupils, students, or even ‘our researchers’, that can give the unfortunate reader the slightest idea of what research is all about - apart from the universally accepted idea that research is something you do when you're really good at it, when you've passed lots of exams and even competitions, the big shots, Pasteur and Curie and the Nobel Prize winners and all that? The rest of us, readers or listeners, are just trying to swallow up the knowledge that these great men were willing to consign for the good of humanity, but we're only good enough (if we work hard) to pass our exams at the end of the year, and even then...

How many there are, including among the unfortunate ‘researchers’ themselves, in need of theses or articles, including even among the most ‘learned’, [◊ 5] the most prestigious among us - who therefore have the simplicity to see that ‘research’ is nothing more or less than questioning things, passionately - like a child who wants to know how he or his little sister came into the world. That seeking and finding, that is to say: questioning and listening, is the simplest, most spontaneous thing in the world, something that no one in the world has the privilege of doing. It's a ‘gift’ that we've all received from the cradle - made to express itself and blossom under an infinite number of faces, from one moment to the next and from one person to the next...
When you dare to say such things, you get the same half-sad, half-understanding smiles from everyone, from the dunce who's sure he's a dunce to the scholar who's sure he's a scholar and well above the common man, as if you'd just made a joke that's a bit too big for words, as if you were displaying a naivety stitched together with white thread; It's all very well to spit on no-one, of course - but don't push it - a dunce is a dunce and not Einstein or Picasso!

In the face of such unanimous agreement, I would be hard pressed to insist. Incorrigible as I am, I've lost another opportunity to keep quiet...

No, it is surely no coincidence that, in perfect harmony, instructive or edifying books and manuals of all kinds present ‘Knowledge’ as if it had emerged from the genius of the brains that recorded it for our benefit. Nor can it be said that this is bad faith, even in the rare cases where the author is ‘in the know’ enough to know that this image (which his text cannot fail to suggest) in no way corresponds to reality. In such cases, the presentation may be more than just a collection of results and recipes, it may be infused with a breath of fresh air, animated by a living vision that is sometimes transmitted from the author to the attentive reader. But an unspoken consensus, of considerable force it seems, means that the text does not leave the slightest trace of the work of which it is the product, even when it expresses with lapidary force the sometimes profound vision of things that is one of the real fruits of this work.

To tell the truth, there have been times when I myself have felt the weight of this force, of this silent consensus, during my project to write and publish these Mathematical Reflections. If I try to fathom the tacit form taken by this consensus, or rather the form taken by the resistance within me to my [◊ 6] project, triggered by this consensus, the term ‘indecency’ immediately comes to mind. The consensus, internalised in me I can't say for how long, says to me (and this is the first time I've taken the trouble to draw into the light of day, into the field of my gaze, what it's been muttering to me with some insistence for weeks, if not months): ‘It's indecent to flaunt before others, even publicly, the ups and downs, the messy trial and error around the edges, the “dirty laundry” in short, of a work of discovery. All it does is waste the reader's precious time. What's more, it's going to add up to pages and pages of material that will have to be typeset and printed - what a waste, at the price of scientific printed paper! You really have to be conceited to flaunt things that are of no interest to anyone, as if my own screw-ups were something remarkable - an opportunity to strut your stuff, in short. And even more secretly: ‘It is indecent to publish the notes of such a reflection, as it is really going on, just as it would be indecent to make love in a public place, or to expose or even just leave lying around the blood-stained sheets of the labours of childbirth...’.

The taboo here takes the insidious yet imperious form of the sexual taboo. It is only as I write this introduction that I begin to glimpse its extraordinary force, and the significance of this very extraordinary fact, attesting to that force: that the real process of discovery, so disconcertingly simple, so childlike in its simplicity, is practically nowhere to be seen; that it is silently suppressed, ignored, denied. This is the case even in the relatively innocuous field of scientific discovery, not that of one's willy or anything like that, thank God - a ‘discovery’ in short that can be put into everyone's hands, and which (one might think) has nothing to hide...
If I wanted to follow the ‘thread’ that runs through here - a thread that is by no means tenuous, but very thick and strong - it would surely take me much further than the few hundred pages of homological-homotopical algebra that I will eventually finish and deliver to the printer.

​ (4) Infallibility(of others) and contempt(of oneself)

It was definitely an understatement, when at times I cautiously observed that ‘my style of expression’ had changed, even suggesting that there was nothing surprising about it: You understand, when you haven't written for thirteen years, it's not the same as before, the ‘style of expression’ must change, inevitably... The difference is that before I ‘expressed myself’ (sic) like everyone else: I did the work, then I did it again in reverse, carefully erasing all the erasures. Along the way, I'd make new cross-outs, turning the whole thing upside down, sometimes worse than the first draft. So it had to be done again - sometimes three or even four times, until everything was perfect. Not only were there no dodgy corners or sweepings surreptitiously shoved under a suitable piece of furniture (I've never liked sweepings in corners, as long as you take the trouble to sweep); above all, when I read the final text, the admittedly flattering impression I got (as with any other scientific text) was that the author (in this case, my modest self) was infallibility incarnate. Infallibly, he came across ‘the’ right concepts, then ‘the’ right statements, with a well-oiled engine humming along, with demonstrations that ‘fell’ with a dull noise, each at exactly the right moment!

Judge for yourself the effect on an unsuspecting reader, let's say a secondary school pupil learning the Pythagorean theorem or equations of the second degree, or even one of my colleagues in so-called ‘higher’ education or research institutions (hear, hear!) shouting (let's say) about reading an article by a prestigious colleague! As this kind of experience is repeated hundreds, thousands of times over the course of a pupil's, or even a student's or researcher's, life, amplified by the appropriate concert in the family as well as in all the media in all the countries of the world, the effect is what you might expect. You can see it in yourself and in others, if you take the trouble to pay attention: it's the intimate conviction of your own worthlessness, in contrast to the competence and importance of people ‘who know’ and people ‘who do’.

This intimate conviction is sometimes compensated for, but in no way resolved or defused, by the development of an ability to memorise things that are not understood, or even by that of a certain operative skill: multiplying matrices, ‘putting together’ a French composition using ‘thesis’ and ‘antithesis’... It's the ability in short of the parrot or the learned monkey, more prized these days than it ever was, sanctioned by coveted diplomas, rewarded by comfortable careers.’ [◊ 8] But even those who are sewn up with diplomas and well-off, perhaps covered in honours, are not fooled, deep down, by these false signs of importance, of ‘value’. Nor even the rarer person who has invested his or her all in the development of some genuine gift, and who in his or her professional life has been able to give his or her all and do creative work - he or she is not convinced, deep down, by the glitter of his or her fame, with which he or she often wants to give the impression to himself or herself and to others. The same unexamined doubt lives in both of them, just as it does in the first dunce who comes along, the same conviction that perhaps they will never dare to acknowledge.

It is this doubt, this intimate unspoken conviction, that drives both of them to constantly surpass themselves in the accumulation of honours or works, and to project onto others (above all onto those over whom they have some power...) this contempt for themselves that gnaws at them in secret - in an impossible attempt to escape it, by accumulating ‘proof’ of their superiority over others (2).

II THE DREAM AND THE DREAMER

Contents

  1. The forbidden dream
  2. The Dreamer
  3. The Galois legacy
  4. Dream and demonstration

February 1984

​ (5) The forbidden dream

I'm taking the opportunity of a three-month break in the writing of the Pursuit of Fields(Poursuite des champs — translator's note) to pick up the Introduction where I left off last June. I've just reread it carefully, more than six months later, and added a few subtitles.

When I wrote this Introduction, I was well aware that this type of reflection could not fail to give rise to many ‘misunderstandings’ - and it would be pointless to try to anticipate them, which would simply amount to accumulating others on top of the first! The only thing I would add in this connection is that I have no intention of waging war against the scientific writing style that has been established by millennia of usage, which I myself have practised assiduously for more than twenty years of my life, and taught to my students as an essential part of the mathematician's profession. Rightly or wrongly, I still regard it as such and continue to teach it. I'm probably even old-fashioned in my insistence on a job well done, hand-stitched from start to finish, with no mercy given to dark corners. If I've had to put water in my wine over the last ten years or so, it's by necessity! For me, ‘writing [◊ 9] in form’ remains an important stage in mathematical work, both as an instrument of discovery, for testing and deepening an understanding of things that would otherwise remain approximate and fragmentary, and as a means of communicating such an understanding. From a didactic point of view, the rigorous method of exposition, the deductive method, which in no way excludes the possibility of painting vast pictures, offers obvious advantages in terms of conciseness and ease of reference. These are real advantages, and significant ones, when it comes to presentations aimed at mathematicians, let's say, and more particularly mathematicians who are already sufficiently familiar with some of the ins and outs of the subject being dealt with, or others very close to it.

These advantages, on the other hand, become entirely illusory for a presentation aimed at children, young people or adults who are not at all ‘in the loop’ beforehand, whose interest is already awakening and who, moreover, are more often than not (and will remain, and for good reason...) in total ignorance of what the real process of discovery work is all about. Readers, to put it better, who are unaware of the very existence of such a work, within everyone's reach with curiosity and common sense - the work from which our intellectual knowledge of the things of the Universe is constantly born and reborn, including that which is expressed in imposing works such as Euclid's Elements, or Darwin's Origin of Species. Complete ignorance of the existence and nature of such work is almost universal, including among teachers at all levels of education, from primary school teachers to university professors. This is an extraordinary fact, which first came to my attention during the reflection I began last year with the first part of this Introduction, at the same time as I was glimpsing the deep roots of this puzzling fact...

Even though it is aimed at readers who are perfectly ‘in the know’ from every point of view, there is still one important thing that the ‘de rigueur’ mode of presentation refuses to communicate. It's also something that's totally frowned upon in the circles of serious people, like us scientists in particular! I'm talking about dreams. The dream, and the visions that it breathes into us - initially impalpable like it, and often reluctant to take shape. Long years, even a lifetime of intense work may not be enough [◊ 10] to see a dream vision fully manifested, to see it condensed and polished to the hardness and brilliance of a diamond. This is our work, workers by hand or by mind. When the work is finished, or a certain part of it, we present the tangible result in the brightest light we can find, we rejoice in it, and often take pride in it. It is not in this diamond, however, which we have cut at length, that we find what inspired us to cut it. Perhaps we have fashioned a tool of great precision, an efficient tool - but the tool itself is limited, like everything made by the hand of man, even when it seems great to us. A vision, nameless and vague at first, tenuous as a wisp of mist, has guided our hand and kept us bent over the work, without feeling the hours or perhaps the years pass. It was a flap that noiselessly detached itself from a bottomless Sea of mist and half-light... What is boundless in us is Her, this Sea ready to conceive and give birth unceasingly, when our thirst makes Her fertile. The Dream arises from these marriages, like the embryo nestled in the nourishing womb, awaiting the obscure labours that will lead it to a second birth, in the light of day.

Woe betide a world where dreams are scorned - it's also a world where what's deep inside us is scorned. I don't know whether any other culture before ours - the culture of television, computers and transcontinental rockets - has professed such contempt. It must be one of the many ways in which we distinguish ourselves from our predecessors, whom we have so radically supplanted, eliminated as it were from the face of the planet. I know of no other culture where the dream is not respected, where its deep roots are not felt by all and recognised. And is there any major work in the life of a person or a people that was not born of a dream and nurtured by a dream before coming to light? In our country (or should we say everywhere?) respect for dreams is called ‘superstition’, and it's well known that our psychologists and psychiatrists have taken the measure of dreams in length, breadth and depth - hardly enough to fill the memory of a small computer, surely. It's also true that no-one ‘back home’ knows how to light a fire, or dares to stand in their own home and watch their child being born, or their mother or father dying - there are clinics and hospitals for that. Thank God... Our world, so proud of its power in atomic megatons and the quantity of information stored in its libraries [◊ 11] and computers, is undoubtedly also the one in which the powerlessness of each individual, that fear and contempt for the simple and essential things in life, has reached its peak.

Fortunately, dreams, like the original sex drive in even the most repressive society, have a way of enduring! Superstition or no superstition, dreams continue to obstinately whisper knowledge that our waking minds are too heavy or too faint-hearted to grasp, and to give life and wings to the projects they have inspired.

I suggested earlier that dreams are often reluctant to take shape, but that's just an appearance, and doesn't really go to the heart of the matter. The ‘reticence’ is more likely to come from our waking mind, in its ordinary ‘state’ - and even then, the term ‘reticence’ is a euphemism! Rather, it's a deep-seated mistrust that covers up an ancestral fear - the fear of knowing. Speaking of dreams in the true sense of the term, this fear is all the more powerful, all the more effective as a screen, because the message of the dream touches us more closely, because it carries with it the threat of a profound transformation of our person, if by chance it were to be heard. But we have to believe that this mistrust is present and effective even in the relatively harmless case of the mathematical ‘dream’; so much so that all dreams seem to be banned not only from texts (I don't know of any where there is any trace of them), but also from discussions between colleagues, in small groups, or even one-to-one.

The reason for this apparent absence, this conspiracy of silence, is certainly not that the mathematical dream does not exist or no longer exists - our science would then have become sterile, which is by no means the case; surely the reason for this apparent absence, this conspiracy of silence, is very closely linked to that other consensus - that of carefully erasing all trace and all mention of the work through which our knowledge of the world is discovered and renewed. Or rather, it is one and the same silence that surrounds both the dream and the work it inspires and nourishes. So much so that the very term ‘mathematical dream’ will seem nonsensical to many of us, who are so often driven by push-button clichés rather than the direct experience we can have of a very simple, everyday, important reality.

​ (6) The Dreamer

[◊ 12] In fact, I know from experience that when the mind is eager to get to know it, instead of running away from it (or approaching it with a patented grid in hand, which amounts to the same thing), the dream is in no way reluctant ‘to take shape’ - to allow itself to be described delicately and to deliver its message, which is always simple, never silly, and sometimes deeply moving. On the contrary, the Dreamer in us is an incomparable master at finding, or creating from scratch, from one occasion to the next, the language best suited to circumventing our fears and shaking our torpor, with theatrical means varying infinitely, from the absence of any visual or sensory element whatsoever to the most breathtaking stagings. When He shows up, it's not to shy away, but to encourage us (almost always to no avail, but His benevolence never wearies...) to get out of ourselves, out of the heaviness in which He sees us trapped, and which He sometimes takes pleasure in parodying in comical colours. Lending an ear to the Dreamer within us means communicating with ourselves, against the powerful barriers that would like to prevent us from doing so at all costs.

But who can do more can do less. If we can communicate with ourselves through dreams, revealing ourselves to ourselves, then surely it must be possible, in an equally simple way, to communicate to others the by no means intimate message of mathematical dreams, let's say, which do not bring into play forces of resistance of comparable power. And to tell the truth, what else have I done in my past as a mathematician, apart from following, ‘dreaming’ to the end, until their most manifest, most solid, indisputable manifestation, the shreds of dream detaching themselves one by one from a heavy, dense fabric of mists? And how many times did I tremble with impatience at my own obstinacy in jealously polishing to the last facet each precious or semi-precious stone in which my dreams were condensed - rather than following a deeper impulse: that of following the multiform arcana of the mother fabric - to the undecided borders of the dream and its patent incarnation, ‘publishable’ in short, according to the canons in force! In fact, I was about to follow that impulse, to embark on a work of ‘mathematical science fiction’, ‘a kind of daydream’ about a theory of ‘patterns’ that remained purely hypothetical at the time - and has remained so to this day, and for good reason, for want of another ‘daydreamer’ to embark on this adventure. This was towards the end of the 1960s, when my life [◊ 13] (without my having the slightest inkling of it) was about to take a completely different turn, which for a decade or so would relegate my mathematical passion to a marginal, if not disowned, position.

But all things considered, In the pursuit of fields, this first publication after fourteen years of silence, is very much in the spirit of that ‘waking dream’ which was never written, and of which it seems to have been the provisional successor. Admittedly, the themes of these two dreams are as dissimilar, at least at first sight, as it is possible for two mathematical themes to be; not to mention that the first, that of motifs, would seem to be situated on the horizon of what could be ‘feasible’ with the means at hand, whereas the second, the famous ‘fields’ and the like, appear to be entirely within reach. These are dissimilarities that could be called fortuitous or accidental, and which will perhaps disappear much sooner than we expect (3). They have relatively little impact, it seems to me, on the kind of work to which either theme can give rise, as long as it is precisely a matter of ‘daydreaming’, or, to put it in less provocative terms: of continuing the work of conceptual rough-casting until an overall vision of sufficient coherence and precision is achieved, to bring about the more or less complete conviction that the vision does indeed correspond, essentially, to the reality of things. In the case of the theme developed in this book, this should mean, more or less, that the detailed verification of the validity of this vision becomes a matter of pure craft. This can certainly require a considerable amount of work, with its share of astuteness and imagination, and no doubt also unexpected twists and perspectives, which will make it something other than a purely routine task (a ‘long exercise’, as André Weil would say).

It's the kind of work, in short, that I've done over and over again in the past, that I have at my fingertips and that it's therefore pointless for me to do again in the years that are still ahead of me. Insofar as I am once again investing myself in mathematical work, it is on the fringes of the ‘daydream’ that my energy will surely be best spent. In this choice, it is not a concern for profitability that inspires me (assuming that such a concern could inspire anyone), but a dream, or dreams. If this new impetus within me is to prove a source of strength, it will have been drawn from the dream!

​ (7) The Galois legacy

[◊ 14] It would seem that of all the natural sciences, it is only in mathematics that what I have called ‘dreaming’, or ‘daydreaming’, is subject to an apparently absolute ban, more than two thousand years old. In the other sciences, including reputedly ‘exact’ sciences such as physics, dreaming is at least tolerated, if not encouraged (depending on the era), under admittedly more ‘outlandish’ names such as : ‘Speculations’, “hypotheses” (such as the famous “atomic hypothesis”, the result of a dream, excuse me, a speculation by Democritus), “theories”... The transition from the status of dream-which-dare-not-say-its-name to that of “scientific truth” is made by insensible degrees, by a consensus that gradually broadens. In mathematics, on the other hand, it is almost always (these days at least) a sudden transformation, by virtue of the magic wand of a demonstration (4). At a time when the notion of mathematical definition and demonstration was not, as it is today, clear and the object of a (more or less) general consensus, there were nevertheless some visibly important notions that had an ambiguous existence - such as that of a ‘negative’ number (rejected by Pascal) or that of an ‘imaginary’ number. This ambiguity is reflected in the language still in use today.

The gradual clarification of the notions of definition, statement, demonstration and mathematical theory has been very salutary in this respect. It has made us aware of all the power of the tools, however childishly simple, that we have at our disposal to formulate with perfect precision the very things that might have seemed unformulate - by virtue of a sufficiently rigorous use of everyday language, more or less. If there is one thing that has fascinated me about mathematics since I was a child, it is precisely this power to capture in words, and to express perfectly, the essence of those mathematical things that at first sight appear in such elusive or mysterious form that they seem beyond words...

However, an unfortunate psychological side-effect of this power, of the resources offered by perfect precision and demonstration, is that they have further accentuated the traditional taboo against the ‘mathematical dream’; that is to say, against anything that does not present itself in the conventional guise of precision (even at the expense of a wider vision), guaranteed ‘in good taste’ by demonstrations in form, or if not (and increasingly so these days...) by sketches of demonstrations, supposed to be able to be put into form. ) by sketches of demonstrations, supposed to be able to take shape. Occasional conjectures are tolerated at a pinch, provided that they satisfy the conditions of precision of a questionnaire, where the only answers allowed would be ‘yes’ or ‘no’. (And, needless to say, on condition that the person who allows himself to do so is well known in the mathematical world). To my knowledge, there has never been an example of the development, on an ‘experimental’ basis, of a mathematical theory that is explicitly conjectural in its essential parts. It's true that, according to modern standards, the entire calculus of the ‘infinitely small’ developed from the seventeenth century onwards, which has since become the differential and integral calculus, would seem like a daydream, which was finally transformed into serious mathematics only two centuries later, by the magic wand of Cauchy. This reminds me of the daydream of Évariste Galois, who had no luck with this same Cauchy; but this time it only took less than a hundred years for another wand, this time from Jordan (if I remember correctly), to give this dream the right to be cited, renamed ‘Galois theory’ for the occasion.

The conclusion that emerges from all this, and which is not to the advantage of ‘Mathematics 1984’, is that it is fortunate that people like Newton, Leibnitz, Galois (and I'm sure there are many more, as I'm not well versed in history...) were not encumbered by our current canons, at a time when they were content to discover without taking the time to canonise!

The example of Galois, who came along without my calling him, strikes a chord with me. I seem to remember that a feeling of fraternal sympathy for him was awakened the first time I heard about him and his strange destiny, when I was still a pupil or student, I think. Like him, I felt a passion for mathematics - and like him I felt like an outsider, a stranger in the ‘beautiful world’ that (it seemed to me) had rejected him. Yet I myself ended up being part of this beautiful world, only to leave it one day, with no regrets... This somewhat forgotten affinity reappeared to me quite recently and in a completely new light, when I was writing the Outline of a program (to coincide with my application to become a researcher at the Centre national de la recherche scientifique). This report is mainly devoted to an outline of my main themes [◊ 16] of reflection over the last ten years or so. Of all these themes, the one that fascinates me the most, and which I intend to develop especially in the next few years, is the very type of mathematical dream, which moreover joins the ‘dream of patterns’, to which it provides a new approach. In writing this outline, I was reminded of the longest mathematical reflection I have pursued in one go in the last fourteen years. It lasted from January to June 1981, and I called it ‘The long walk through Galois theory’. One thing led to another and I realised that the daydream I had been pursuing sporadically for several years, which had come to be known as ‘Anabelian algebraic geometry’, was nothing other than a continuation, ‘an ultimate culmination of Galois's theory, and undoubtedly in the spirit of Galois’.

When this continuity occurred to me, as I was writing the passage from which the quoted line is taken, a joy came over me that has not dissipated. It was one of the rewards of working in complete solitude. Its appearance was as unexpected as the more than fresh welcome I had received from two or three colleagues and old friends who were well ‘in the know’, one of whom was my pupil, and to whom I had had the opportunity to talk, still ‘on the spot’ and in the joy of my heart, about these things I was in the process of discovering...

It reminds me that to take up Galois' legacy today is surely also to accept the risk of the solitude that was his in his time. Perhaps times are changing less than we think, but often this ‘risk’ does not feel like a threat to me. Although I am sometimes saddened and frustrated by the indifference or disdain of those I have loved, never for many years has solitude, mathematical or otherwise, weighed on me. If there's one faithful friend that I always long to find again when I leave her, it's her!

​ (8) Dream and demonstration

But let's get back to dreams, and the prohibition that has dogged mathematics for thousands of years. This is perhaps the most inveterate of all the preconceptions(a priori — translator's note), often implicit and rooted in habit, decreeing that one thing is ‘maths’ and another is not. It took millennia for things as childish and ubiquitous as the symmetry groups of certain geometric figures, the topological forms of others, the number zero, and sets to find admission into the [◊ 17] sanctuary! When I talk to students about the topology of a sphere, and the shapes that can be deduced from a sphere by adding coves - things that don't surprise young children, but baffle them because they think they know what ‘maths’ is - the first spontaneous echo I get is: but that's not maths! Maths, of course, is the Pythagorean theorem, the heights of a triangle and second-degree polynomials... These students are no more stupid than you or I. They react as all the world's mathematicians have reacted from time immemorial to the present day, with the exception of people like Pythagoras or Riemann and maybe five or six others. Even Poincaré, who wasn't the first to come along, managed to prove with a well-felt philosophical A plus B that infinite sets weren't maths! Surely there must have been a time when triangles and squares weren't maths - they were drawings that children or craftsmen traced in the sand or clay of vases, not to be confused...

This fundamental inertia of the mind, suffocated by its ‘knowledge’, is certainly not unique to mathematicians. I'm getting a bit off track here: the ban on the mathematical dream, and through it, on anything that doesn't have the usual appearance of a finished product, ready for consumption. The little I have learnt about the other natural sciences is enough to make me realise that a ban of similar rigour would have condemned them to sterility, or to a tortoise's progress, rather like in the Middle Ages when there was no question of dehornifying the letter of the Holy Scriptures. But I am also well aware that the deep source of discovery, just like the process of discovery in all its essential aspects, is the same in mathematics as in any other region or thing in the Universe that our body and mind can experience. To banish the dream, is to banish the source - to condemn it to an occult existence.
And I am also well aware, from an experience that has not wavered since my first and juvenile love affair with mathematics, that in the unfolding of a vast or profound vision of mathematical things, it is this unfolding of a vision and an understanding, this progressive penetration, that constantly precedes the demonstration, that makes it possible and gives it meaning. When a situation, from the humblest to the most vast, has been understood in its essential aspects, the demonstration of what is understood (and of the rest) falls like ripe fruit. Whereas the demonstration plucked [◊ 18] like a still green fruit from the tree of knowledge leaves an aftertaste of dissatisfaction, a frustration of our thirst, by no means appeased. Two or three times in my life as a mathematician I have had to resort, for want of anything better, to plucking the fruit rather than picking it. I'm not saying that I did wrong, or that I regret it. But what I did best and what I liked best, I took willingly, not by force. If mathematics has given me profuse joy and continues to fascinate me in my mature years, it is not because of the demonstrations that I have been able to wring from it, but because of the inexhaustible mystery and perfect harmony that I sense in it, always ready to reveal itself to a loving hand and gaze.

III BIRTH OF FEAR

Contents

  1. The welcome stranger
  2. The 'mathematical community': fiction and reality
  3. Metting with Claude Chevalley——or freedom and good feelings
  4. Merit and contempt
  5. strength and thickness
  6. Birth of fear
  7. Harvest and sowing

​ (9) The welcome stranger

I think the time has come to say something about my relationship with the world of mathematicians. This is quite different from my relationship with mathematics. It existed and was strong from an early age, long before I even realised that there was a world and a milieu of mathematicians. A complex world, with its learned societies, its periodicals, its meetings, colloquia and congresses, its prima donnas and its taskmasters, its power structure, its grey eminences, and the no less grey mass of those who can be tamed and bribed, in need of a thesis or articles, and also those, rarer still, who are rich in means and ideas and come up against closed doors, desperate to find the support of one of those powerful, pressurised and feared men who have that magic power: to get an article published...

I discovered the existence of a mathematical world when I arrived in Paris in 1948, at the age of twenty, with a Bachelor of Science degree from the University of Montpellier in my meagre suitcase, and a tightly-written manuscript on both sides, with no margins (paper was expensive!), representing three years of solitary reflection on what (I later learned) was then well known as the ‘theory of measurement’ or the ‘Lebesgue integral’. As I had never met anyone else, I thought, until the day I arrived in Paris, that I was the only person in the world ‘doing maths’, the only mathematician. (It was the same thing for me, and remains so to this day). I had juggled with what I called ‘measurable’ sets (without ever having come across a set that wasn't measurable...) and with convergence almost everywhere, but I didn't know what a topological space was. I was a bit lost in a dozen or so non-equivalent notions of ‘abstract space’ and compactness, which I had picked up in a little booklet (by someone called Appert, I think, in Scientific and industrial news(Actualités scientifiques et industrielles — translator's note)), which I had stumbled across somehow. I'd never heard before, in a mathematical context at least, such strange or barbaric words as group, field, ring, module, complex, homology (and so on!), which suddenly, without warning, came crashing down on me all at once. It was a rude shock!

If I ‘survived’ this shock, and went on to do maths and even make it my profession, it's because in those distant days the mathematical world was hardly what it has become since. It's also possible that I was lucky enough to land in a more welcoming corner of this unsuspected world. I had a vague recommendation from one of my professors at the Faculty of Montpellier, Monsieur Soula (no more than his colleagues, he had not seen me often in his classes!), who had been a pupil of Cartan (father or son, I couldn't say). As Élie Cartan was already ‘out of the game’ by then, his son, Henri Cartan, was the first ‘fellow’ I had the pleasure of meeting. I had no idea how auspicious that was! I was greeted by him with the kindly courtesy that distinguishes him, well known to the generations of normaliens who were lucky enough to have had their first experience with him. He must not have realised the extent of my ignorance, judging by the advice he gave me at the time to guide my studies. Be that as it may, his benevolence was clearly aimed at the person, not at any potential baggage or gifts, nor (later on) at a reputation or notoriety...

In the year that followed, I was the host of one of Cartan's lectures at the ‘École’ (on the differential form of varieties), to which I clung firmly; I was also the host of the ‘Cartan seminar’, witnessing in amazement the discussions between him and Serre, with their ‘Spectral Sequences’ (brr!) and drawings (called ‘diagrams’) full of arrows covering the whole board. Those were the heroic days of the theory of ‘beams’, ‘carapaces’ and a whole arsenal whose meaning totally escaped me, even though I was doing my best to swallow definitions and statements and check demonstrations. At the Cartan seminar there were also regular appearances by Chevalley and Weil, and on the days of the Bourbaki seminars (which brought together [◊ 20] at least twenty or thirty participants and listeners), you could see the other members of the famous Bourbaki gang: Dieudonné, Schwartz, Godement and Delsarte. They were all on a first-name basis, spoke the same language that almost totally escaped me, smoked a lot and laughed a lot. The only thing missing was the crates of beer to complete the atmosphere - that was replaced by chalk and sponges. It was a completely different atmosphere from Leray's lectures at the Collège de France (on Schauder's theory of topological degree in infinite-dimensional spaces, poor me!), which I went to listen to on Cartan's advice. I had gone to see Monsieur Leray at the Collège de France to ask him (if I remember correctly) what his lecture would be about. I don't remember what explanations he gave me, or whether I understood anything - only that there too I felt a kindly welcome, addressed to the first stranger who came along. It was this and nothing else, surely, that made me go to this course and bravely stick with it, as I had done at the Cartan seminar, even though the meaning of what Leray was explaining escaped me almost completely.

The strange thing was that in this world where I was a newcomer and whose language I hardly understood, let alone spoke, I didn't feel like a stranger. Although I hardly ever had the opportunity to talk (and for good reason!) with one of those cheerful fellows like Weil or Dieudonné, or with one of those more distinguished gentlemen like Cartan, Leray or Chevalley, I nevertheless felt accepted, I would almost say: one of them. I can't recall a single occasion when I was treated with condescension by one of these men, or when my thirst for knowledge, and later, once again, my joy in discovery, was rejected by smugness or disdain (5). Had it not been so, I would not have ‘become a mathematician’ as they say - I would have chosen another profession, where I could give my all without having to face scorn...

Even though I was ‘objectively’ a stranger to this world, just as I was a stranger to France, a link united me to these men from another background, another culture, another destiny: a shared passion. I doubt that in that crucial year when I was discovering the world of mathematicians, any of them, not even Cartan, of whom I was a bit of a pupil but who had many others (and some of them not so out of touch!), perceived in me the same passion that [◊ 21] inhabited them. For them, I must have been one of a mass of course and seminar listeners, taking notes and obviously not quite up to speed. If perhaps I stood out in any way from the other listeners, it was that I wasn't afraid to ask questions, which more often than not had to denote above all my phenomenal ignorance of both language and mathematical things. The answers could be brief, or even astonished, but never did the bemused eccentric that I was then encounter a rebuff, a ‘putting in my place’, either in the informal environment of the Bourbaki group, or in the more austere setting of the Leray course at the Collège de France. In those years, ever since I arrived in Paris with a letter to Élie Cartan in my pocket, I've never had the impression of finding myself in front of a clan, a closed or even hostile world. If I've ever experienced this inner contraction in the face of contempt, it wasn't in that world; at least not in those days. Respect for the individual was part of the air I breathed there. You didn't have to earn respect, you didn't have to prove yourself before you were accepted, and you didn't have to be treated kindly. Strangely enough perhaps, it was enough to be a person, to have a human face.

​ (10) The 'mathematical community': fiction and reality

So it's hardly surprising that, perhaps from that year onwards, and in any case more and more clearly over the years that followed, I felt part of this world, to which I was happy to refer under the name, charged with meaning for me, of ‘mathematical community’. Before writing these lines, I never had the opportunity to examine the meaning I gave to this name, even though I identified with this ‘community’ to a large extent. It's clear now that this community represented for me nothing more and nothing less than a kind of ideal extension, in space and in time, of the benevolent world that had welcomed me and accepted me as one of their own; a world, moreover, to which I was linked by one of the great passions that dominated my life.
This ‘community’, with which I gradually became identified, was not an entirely fictitious extrapolation of the mathematical environment that had initially welcomed me. The initial milieu expanded little by little, I mean: the circle of mathematicians I was led to frequent regularly, driven by common themes of interest and personal affinities, went from strength to strength in the ten or twenty years following that first contact. In [◊ 22] concrete terms, it was the circle of colleagues and friends, or rather this concentric structure ranging from the colleagues with whom I was most closely associated (first Dieudonné, Schwartz, Godement, later especially Serre, later still people like Andreotti, Lang, Tate, Zariski, Hironaka, Mumford, Bott, Mike Artin, not to mention the people in the Bourbaki group, which was also gradually expanding, and the students who came to me from the 1960s onwards... ), and other colleagues whom I had met here and there and to whom I was linked more or less closely by more or less strong affinities - it was this microcosm, formed by chance encounters and affinities, which represented the concrete content of this name charged with warmth and resonance for me: the mathematical community. When I identified with it as a living, warm entity, it was in fact this microcosm that I was identifying with.

It was only after the ‘great turning point’ of 1970, the first awakening I should say, that I realised that this cosy, friendly microcosm represented only a very small part of the ‘mathematical world’, and that the traits I liked to attribute to this world, which I continued to ignore and had never thought of taking an interest in, were fictitious traits.

In the course of these twenty-two years, this microcosm itself had changed its face, in a surrounding world that was also changing. I too had certainly changed over the years, without realising it, just like the world around me. I don't know whether my friends and colleagues were more aware of this change than I was, in the world around them, in their own microcosm, and in themselves. Nor can I say when or how this strange change came about - it undoubtedly came about insidiously, in wolf's steps: the man of notoriety was feared. I myself was feared - if not by my students, then by my friends, or by those who knew me personally, at least by those who knew me only through notoriety, and who did not feel themselves protected by comparable notoriety.

I only became aware of the fear that is rife in the mathematical world (and just as much, if not more so, in other scientific circles) in the aftermath of my ‘awakening’ nearly fifteen years ago. During the fifteen years that had preceded, gradually and without suspecting it, I had entered the role of the ‘big boss’, in the world of the mathematical Who's who. Without [◊ 23] suspecting it either, I was a prisoner of this role, which isolated me from everyone except a few ‘peers’ and a few students (and even then...) who decidedly ‘wanted it’. It was only when I got out of that role that at least some of the fear surrounding it fell away. Tongues were loosened that had been silent before me for years.

The testimony they brought me was not only that of fear. It was also one of contempt. Especially the contempt of those in power for others, a contempt that creates and feeds fear.

I didn't have much experience of fear, but I did have experience of contempt, at a time when the person and the life of a person didn't carry much weight. It had pleased me to forget the time of contempt, and now it had come back to haunt me! Perhaps it had never stopped, when I had been content simply to change the world (as it seemed to me), to look elsewhere, or simply: to pretend not to see or hear anything, apart from the fascinating and interminable mathematical discussions? These were the days when I finally came to terms with the fact that contempt was rampant all around me, in the world I had chosen as my own, with which I had identified, which had had my backing and which had pampered me.

​ (11) Metting with Claude Chevalley——or freedom and good feelings

Perhaps the preceding lines give the impression that I was overwhelmed by the testimonies that began pouring in almost overnight. But this was not the case. These testimonies were recorded on a superficial level. They were simply added to other facts that I had just learned, or that I knew about but had avoided paying attention to until then. Today, I would express the lesson I learned then as follows: ‘Scientists', from the most illustrious to the most obscure, are people just like everyone else! I had allowed myself to imagine that ‘we’ were something better, that we had something extra - it took me a good year or two to get rid of that delusion, which was decidedly tenacious!

Among the friends who helped me to do this, only one was part of the milieu I had just left with no desire to return (6). That was Claude Chevalley. Although he didn't make speeches and wasn't interested in mine, I think I can say that I learnt more important and more hidden things from him than those I've just mentioned. In the days when I used to see him quite [◊ 24] regularly (the days of the ‘Surviving’ group, which he had joined with mixed conviction), he often baffled me. I couldn't say how, but I felt that he had a knowledge that I didn't have, an understanding of certain essential things that are surely very simple, that can be expressed in simple words certainly, but without the understanding ‘passing’ from one to the other. I realise now that there was a difference in maturity between him and me, which meant that I often felt at odds with him, in a kind of dialogue of the deaf that wasn't due to a lack of mutual sympathy or esteem. Although he didn't express himself in these terms (as far as I can remember), it must have been clear to him that the ‘questioning’ (about the ‘social role of the scientist’, about science, etc.) that I was coming to at the time, either on my own or through the logic of joint reflection and activity within the ‘Survivre’ group (which later became ‘Survivre et vivre’) - that this questioning was basically superficial. They concerned the world in which I lived, certainly, and even the role I played in it - but they didn't really involve me in any profound way. My view of myself, during those heady years, didn't change one bit. It wasn't then that I began to get to know myself. It was only six years later that, for the first time in my life, I got rid of a persistent illusion, not about other people or the world around me, but about myself. It was another awakening, more far-reaching than the first that had prepared it. It was one of the first in a whole ‘cascade’ of successive awakenings, which I hope will continue in the years that remain to me.

I don't recall Chevalley ever alluding to self-knowledge, or ‘self-discovery’ to put it better. In retrospect, however, it's clear that he must have started getting to know himself a long time ago. He sometimes spoke about himself, just a few words on the occasion of this or that, with disconcerting simplicity. He was one of the two or three people I never heard come up with a cliché. He spoke very little, and what he said expressed, not ideas that he had adopted and made his own, but a personal perception and understanding of things. That's why I'm sure he often disconcerted me, even when we were still meeting in the Bourbaki group. What he said often upset ways of seeing that were [◊ 25] dear to me, and which for that reason I considered to be ‘true’. There was an inner autonomy in him that I lacked, and which I began to perceive obscurely at the time of ‘Surviving and Living’. This autonomy is not a matter of intellect or discourse. It's not something you can ‘adopt’, like ideas, points of view, etc. The idea would never have occurred to me. Fortunately, the idea would never have occurred to me to want to ‘make my own’ this autonomy perceived in another person. I had to find my own autonomy. That also meant learning (or relearning) to be myself. But in those years, I had no idea of my lack of maturity, of inner autonomy. If I ended up discovering it, surely the meeting with Chevalley was one of the factors that silently worked within me, at a time when I was embarking on major projects. It wasn't speeches or words that sowed that seed. To sow it, it was enough for a person I met along the way to dispense with speeches and just be himself.

It seems to me that in the early 1970s, when we met regularly to publish the Survivre et vivre newsletter, Chevalley was trying, without insisting, to communicate a message to me that I was then too clumsy to grasp, or too wrapped up in my militant tasks. I was obscurely aware that he had something to teach me about freedom - about inner freedom. Whereas I tended to operate on the basis of grand moral principles and had started to blow that trumpet from the very first issues of Survivre, as a matter of course, he had a particular aversion to moralistic discourse. I think that was the thing that most baffled me about him in the early days of Survivre. For him, such discourse was just an attempt at constraint, superimposed on a multitude of other external constraints stifling the individual. Of course, you could spend a lifetime discussing the pros and cons of such a viewpoint. It totally overturned mine, which (as you can imagine) was driven by the noblest and most generous feelings. It was incomprehensible to me that Chevalley, for whom I had the highest esteem and with whom I felt a bit like a comrade in arms, should take such malicious pleasure in not sharing these feelings! I didn't understand that the truth, the reality of things, is not a question of good feelings, points of view or preferences. Chevalley saw one thing, something simple [◊ 26] and real, and I didn't see it. It wasn't that he had read it somewhere; there's nothing in common between seeing something, and reading something about it. You can read a text with your hands (in Braille script) or with your ears (if someone reads it to you), but you can only see the thing itself with your own eyes. I don't think Chevalley had better eyes than me. But he used them, and I didn't. I was too caught up in my good feelings and everything else to have the leisure to look at the effect of my good feelings and principles on myself and on others, starting with my own children.

He must have realised that I often didn't use my eyes, that I didn't even want to. It's strange that he never let me know. Or did he, without my hearing? Or did he refrain from doing so, judging it to be a wasted effort? Or perhaps the idea didn't even occur to him - it was my business after all, not his, whether I used my eyes or not!

​ (12) Merit and contempt

I would like to take a closer look, in the light of my own limited experience, at when and how contempt took hold in the world of mathematicians, and more particularly in that ‘microcosm’ of colleagues, friends and students that had become like my second home. And at the same time, to see what part I played in this transformation.

I think I can say, without reservation, that in 1948-1949, in the circle of mathematicians I mentioned earlier (whose centre for me was the original Bourbaki group), I did not encounter the slightest trace of contempt, or simply disdain or condescension, towards myself or any of the other young people, French or foreign, who had come there to learn the profession of mathematician. The men who played a leading role, because of their position or prestige, such as Leray, Cartan and Weil, were not feared by me, nor I believe by any of my fellow students. Apart from Leray and Cartan, who were very ‘distinguished gentlemen’, it even took me a good while before I realised that each of these fellows who turned up there with no manners, addressing Cartan as a friend and obviously ‘in the know’, was a university professor just like Cartan himself, in no way lived from hand to mouth like me but received what I considered astronomical emoluments, and was moreover a mathematician of international stature and audience.

[27] Following a suggestion from Weil, I spent the next three years in Nancy, which at that time was Bourbaki's headquarters, with Delsarte, Dieudonné, Schwartz, Godement (and a little later also Serre) teaching at the university. There was only a handful of four or five young people there with me (including Lions, Malgrange, Bruhat and Berger, unless I'm mistaken), so there was much less ‘drowning in the crowd’ than in Paris. The atmosphere was even more familiar, everyone knew each other personally, and we were all on first-name terms, I think. When I look back in my memory, however, this is the first and only time I saw a mathematician treat a student with undisguised contempt. The unfortunate fellow had come for the day, from another town, to work with his boss (he was to prepare a doctoral thesis, which he eventually passed with flying colours, and has since acquired a certain notoriety, I believe). I was quite taken aback by the scene. If someone had used that kind of tone with me for even a second, I'd have slammed the door in their face! As it was, I knew the ‘boss’ well, I was even on speaking terms with him, not the student I only knew by sight. My eldest had, in addition to a broad culture (not only mathematical) and an incisive mind, a kind of peremptory authority that impressed me at the time (and for quite a long time afterwards, until the early 1970s). He had a certain hold over me. I don't remember if I asked him a question about his attitude, only the conclusion I drew from the scene: that this unfortunate pupil must really be a loser to deserve to be treated like that - something like that. I didn't say to myself that if the pupil was indeed rubbish, that was a reason to advise him to do something else, and to stop working with him, but in no way to treat him with contempt. I identified with the ‘maths whizzes’ such as this prestigious senior, at the expense of the ‘nobodies’ who were to be despised. So I followed the ready-made path of connivance with contempt, which suited me, by highlighting the fact that I was accepted into the brotherhood of deserving people, those who were good at maths! (7)

Of course, I, no more than anyone else, would have said to myself in no uncertain terms: people who try their hand at maths and don't succeed are fit to be despised! I would have heard someone say something in that water, around that time or any other, and I would have taken it back in fine fashion, sincerely sorry for such phenomenal [◊ 28] spiritual ignorance. The fact is that I was bathed in ambiguity, I was playing on two sides that weren't communicating: on the one hand, fine principles and feelings, on the other: poor guy, you really have to be rubbish to be treated like that (innuendo: it's not to me that this kind of misadventure could happen, that's for sure!).
In the end, it seems to me that the incident I reported, and above all the (apparently harmless) role I played in it, is in fact typical of an ambiguity in me, which followed me throughout my life as a mathematician in the twenty years that followed, and which only dissipated in the aftermath of the ‘awakening’ of 1970 (8), without my clearly detecting it until today, when I'm writing these lines. It's a pity that I didn't realise it at the time. Perhaps the time was not ripe for me. In any case, the evidence I was hearing at the time about the reign of contempt to which I had chosen to turn a blind eye did not implicate me personally, nor indeed any of my colleagues and friends in the part of my beloved microcosm closest to me (9). It was more to the tune of: ah! how sad to have to learn (or: to teach you) such things, who would have thought, you really have to be a bastard (I was going to say: null, sorry!) to treat living beings that way! In the end, it's not so different from the other way around: just replace ‘suck’ with ‘bastard’ and ‘be treated’ with ‘treat’ and you're done! And honour, of course, is safe for the champion of good causes!

What's clear from this is my connivance with attitudes of contempt. At the very least, it goes back to the very early 1950s, in other words to the years after Cartan and his friends had been kindly received. If I didn't ‘see anything’ later on, when contempt was becoming commonplace just about everywhere, it was because I didn't want to see - any more than in this isolated and particularly flagrant case, where you really had to pull out all the stops to pretend not to see or feel anything!

This complicity was in close symbiosis with my new identity, that of a respected member of a group, the group of deserving people, the maths whizzes. I remember feeling particularly satisfied, proud even, that in this world that I had chosen for myself, that had co-opted me, it wasn't social position or even (but no! ) reputation alone counted, even if it had to be deserved - you could be a university professor [◊ 29] or an academician or whatever, but if you were just a mediocre mathematician (poor guy!) you were nothing, what counted was merit alone, deep, original ideas, technical virtuosity, vast visions and all that!

This ideology of merit, with which I had identified unreservedly (even though it remained, of course, implicit and unspoken), still took a heavy toll on me in the aftermath, as I said, of the famous awakening of 1970. I'm not sure that it disappeared without a trace from that moment on. For that to have happened, I would probably have had to detect it clearly in myself, whereas it seems to me that I was mainly denouncing it in others. In fact, it was Chevalley who was one of the first, along with Denis Guedj, whom I also met through Survivre, to draw my attention to this ideology (they called it ‘meritocracy’, or something like that), and the violence and contempt it contained. It was because of this, Chevalley told me (it must have been at the time of our first meeting at his house, about Survivre), that he could no longer stand the atmosphere at Bourbaki and stopped going there. Looking back, I'm convinced that he must have realised that I had indeed been part of that ideology, and perhaps even that traces of it still remained in some corners. But I don't remember him ever suggesting that. Perhaps he preferred to leave it to me to dot the i's and cross the t's, and I waited until now to do so. Better late than never!

​ (13) strength and thickness

It's quite possible that the incident I reported also marks the moment of an inner shift in me, towards a more or less unconditional identification with the brotherhood of merit, at the expense of people considered to be worthless, or simply ‘without genius’ as we would have said a few generations before (this term was no longer in vogue in my day): dull, mediocre people - at best ‘sounding boards’ (as Weil wrote somewhere) for the great ideas of those who really matter... The mere fact that my memory, which so often acts as a gravedigger even for episodes that at the time mobilise considerable psychic energy, retained this one episode, not attached to any other directly related memory, and presenting itself in such an innocuous guise, makes plausible this feeling of a ‘tipping point’ that would then have taken place.’

[◊ 30] In a meditation less than five years ago, I came to realise that this ideology of ‘we, the great and noble minds...’, in a particularly extreme and virulent form, had plagued my mother since childhood, and dominated her relationship with others, whom she liked to look down on from the height of her grandeur with an often disdainful, even contemptuous commiseration. I had unreserved admiration for my parents. The first and only group with which I identified, before the famous ‘mathematical community’, was the family group reduced to my mother, my father and myself, who had had the honour of being recognised by my mother as worthy of having them as parents. In other words, the seeds of contempt must have been sown in me from childhood. The time might be ripe to follow the vicissitudes, through my childhood and my adult life, of these seeds, and the harvests of illusion, isolation and conflict into which some of them have grown. But that's not the place here, where I have a more limited purpose. I think I can say that this attitude of contempt has never in my life taken on a vehemence and destructive force comparable to those I saw in my mother's life (when I took the trouble to look at my parents' lives, twenty-two years after my mother's death, and thirty-seven years after my father's). But now is as good a time as any to take a close look at at least what place this attitude has had in my life as a mathematician.

Before doing so, to put the incident reported in the previous paragraph into its general context, I would like to emphasise that it stands entirely alone among my memories of the 1950s, and even later. Even today, when I note a sometimes disconcerting erosion of certain elementary forms of courtesy and respect for others in the environment that was once mine (10), the direct and undisguised expression of contempt from boss to pupil must be a fairly rare occurrence. As far as the 1950s are concerned, I have very few memories of any fear that might have surrounded a well-known figure, or of a contemptuous or simply disdainful attitude. If I dig into this, I can say that the first time I was received at Dieudonné's home in Nancy, with the delicate kindness he always showed me, I was a little taken aback by the way this refined and affable man talked about his students - all of them morons, I might add! It was a chore to give them lectures, which they obviously didn't understand... After 1970 I heard the echoes [◊ 31] coming from the lecture theatre side, and I knew that Dieudonné was well and truly feared by the students. Yet while he had a reputation for having strong opinions and serving them up with sometimes thunderous frankness, I never saw him behave in a hurtful or humiliating way, including in the presence of colleagues whom he held in low esteem, or at times of his legendary big tantrums, which subsided as quickly and easily as they had arisen.

Without associating myself with the feelings expressed by Dieudonné about his students, I didn't distance myself from his attitude either, presented as the most obvious thing in the world, as almost self-evident from someone who had a passion for mathematics. Thanks to the benevolent authority of my elder brother, this attitude seemed to me to be at least one of the possible attitudes one could reasonably have towards students and teaching tasks.

It seems to me that for Dieudonné and myself, both imbued with the same ideology of merit, its isolating effect was largely neutralised when we found ourselves in front of a real person, whose very presence silently reminded us of more essential realities than those of so-called ‘merit’, and re-established a forgotten link. The same must have been true for most of our colleagues and friends, no less imbued than Dieudonné or myself with the widespread superiority syndrome. This is no doubt still the case today for many of them.

Weil also had a reputation for being feared by his students, and he was the only one in my microcosm, in the 1950s, whose reputation I had the impression was feared even among colleagues of more modest status (or simply temperament). At times, he would display attitudes of unremitting haughtiness, which could disconcert even the most hardened of self-confidence. My susceptibility helped, and once or twice this led to a passing quarrel. I didn't perceive any hint of contempt in his manner or any deliberate intention to hurt or crush; rather, he had the attitude of a spoilt child, taking pleasure (sometimes maliciously) in making people feel uncomfortable, as if to convince himself of the power he wielded. Moreover, he had a truly astonishing ascendancy over the Bourbaki group, which he sometimes gave me the impression of leading by the hand, rather like a kindergarten teacher leading a troop of well-behaved children.

[32] I can only remember one other occasion in the 1950s when I felt a brutal, undisguised expression of contempt. It came from a foreign colleague and friend, about my age. He had uncommon mathematical power. A few years earlier, when this power was already quite apparent, I had been struck by his submission (which seemed to me almost obsequious) to the great professor whose modest assistant he still was. His exceptional abilities soon earned him an international reputation and a key position at a particularly prestigious university. There he reigned over a small army of student assistants, apparently just as absolutely as his boss had reigned over him and his fellow students. When I asked him (if I remember correctly) if he had any students (by which he meant students who did a good job with him), he replied, with an air of false casualness (I'm translating into French): ‘douze pièces! - where ‘pièces’ was the name by which he referred to his students and assistants. It is certainly rare for a mathematician to have so many students at the same time doing research under his direction - and surely my interlocutor was secretly proud of this, which he tried to hide under his careless air, as if to say: ‘Oh, just twelve pieces, not even worth talking about! That must have been around 1959, I already had a good shell so surely, I still gagged! I must have told him straight away one way or another, and I don't think he was angry with me. Perhaps even his relationship with his students wasn't as sinister as his expression might suggest (I didn't get a testimonial from one of his students), and he'd simply been caught up in his childish desire to strut before me in all his glory. Looking back, I can see that this incident must have marked a turning point in our relationship, which had been one of friendship - I sensed in him a kind of fragility, a finesse too, which attracted my affectionate sympathy. These qualities had become blunted, corroded by his position as an important man, admired and feared. After that incident, I still felt uneasy about him - I definitely didn't feel part of the same world as him...

And yet we were part of the same world - and without realising it any more than he did, I was probably getting thicker too. I still have a vivid memory of the Edinburgh International Congress in 1958. Since the previous year, with my work on the Riemann-[◊ 33] Roch theorem, I had been promoted to the big star, and (without my having had to say so to myself in no uncertain terms at the time) I was also one of the stars of the Congress. (I gave a talk there on the vigorous start of schema theory in that same year). Hirzebruch (another star of the day, with his own Riemann-Roch theorem) was giving a keynote speech, in honour of Hodge who was retiring this year. At one point, Hirzebruch suggested that mathematics was made by the work of young people in particular, rather than that of mature mathematicians. This triggered a general outcry of approval in the Congress hall, where young people formed the majority. I was delighted and very much in agreement, of course - I was thirty on the dot, so I could still pass for young, and the world belonged to me! In my enthusiasm, I had to shout out loud and bang my head on the table. I happened to be sitting next to Lady Hodge, the wife of the eminent mathematician who was supposed to be honoured on this occasion, as he was about to retire. She turned to me with wide eyes and said a few words that I can no longer remember - but I must have seen reflected in her astonished eyes the tactless thickness that had just been unleashed on this lady at the end of her life. I felt something then, which the word ‘shame’ perhaps gives a distorted image of - rather a humble truth about who I was then. I didn't have to bang on any more tables that day...

​ (14) Birth of fear

It was around this time I suppose, when (without having sought it) I began to be seen as a star in the mathematical world, that a certain fear must also have begun to surround my person, for a good number of unknown or lesser-known colleagues. I suppose so, without being able to place it in a precise memory, in an image that struck me and became fixed in my memory, like the incident reported earlier (which undoubtedly marked my first encounter with contempt in my adopted environment). It must have happened imperceptibly, without attracting my attention, without manifesting itself in some particular, typical incident that memory would have remembered, in a light perhaps just as deliberately anodyne as that other incident. What I remember ‘en bloc’ from those years of transition is that it was not unusual for people who approached me, whether after my seminar, or during a meeting such as the Bourbaki seminar or some [◊ 34] colloquium or congress, to have to overcome a kind of stage fright, which remained more or less apparent during our discussion, if there was one. When this lasted more than a few minutes, this discomfort more often than not disappeared gradually as we spoke and the conversation became more animated. Occasionally, on rare occasions, the discomfort would persist, to the point of becoming a real obstacle to communication, even at the impersonal level of a mathematical discussion, and I would then feel a confused sense of powerless suffering in front of me, exasperated with itself. I'm talking about all this without really ‘remembering’, as if through a fog that nevertheless gives me back impressions that must have been recorded, and no doubt evacuated as I went along. I wouldn't be able to place in time, other than by supposition, the appearance of this discomfort, the expression of a fear.

I don't believe that this fear emanated from my person and that it was limited to an attitude, to behaviours that would have distinguished me from my colleagues. If that had been the case, it seems to me that I would have ended up receiving echoes of it in the early 1970s, when I stepped out of a role to which I had lent myself until then, the role of the star, the ‘big boss’. It was this role, I think, and not myself, that was surrounded by fear. And this role, it seems to me, with this halo of fear that has nothing in common with respect, did not exist, not yet, in the early 1950s, at least not in the mathematical community that had welcomed me from the very moment I met it, in 1948.

Before this ‘awakening’ in 1970, I wouldn't have thought of describing as ‘fear’ the stage fright and embarrassment that I was sometimes confronted with by colleagues who weren't part of my most familiar environment. I was embarrassed by it myself when it appeared, and did everything I could to dispel it. A remarkable thing, typical of the lack of attention paid to this kind of thing in my beloved microcosm: I can't remember a single time, during the twenty years I was part of this milieu, when the question was raised between a colleague and myself, or by others in front of me! (11) Nor does this ‘fog’ that serves as my memory give me any impression of conscious or unconscious gratification that such situations might have aroused in me. I don't think there were any at the conscious level, but I wouldn't venture to say that I wasn't occasionally touched by them [◊ 35] at the unconscious level, in the early years. If so, it must have been fleeting, without reverberating in any behaviour that would have acted as a fixative for a discomfort. It's certainly not that my fatuity wasn't involved in the role I was playing! But if I invested all my energies in this role, then what motivated my ego was not the ambition to impress the ‘colleague in the row’, but to constantly surpass myself in order to win the ever-renewed esteem of my ‘peers’ - and above all, perhaps, of the elders who had given me credit and accepted me as one of their own even before I had had a chance to prove myself. It seems to me that my inner attitude towards the fear I was the object of, which I tried my best to ignore while at the same time dispelling it as best I could where it manifested itself - that attitude can be considered typical throughout the 1960s in the milieu (the ‘microcosm’) of which I was a part.

The situation has deteriorated considerably in the ten or fifteen years that have passed since then, at least judging by the signs that reach me from time to time from this world, and the situations of which I have been a close witness, even sometimes a co-actor. More than once, among those of my former friends or students who had been dearest to me, I have been confronted with the familiar, unmistakable signs of contempt; with the (seemingly ‘gratuitous’) desire to discourage, humiliate and crush. A wind of contempt blew through this world that had been dear to me. It blows without regard to ‘merit’ or ‘demerit’, burning with its breath the humblest vocations as well as the most beautiful passions. Is there a single one among my companions of yesteryear, each protected by solid walls with ‘his own’, settled (as I once was) in the hushed fear that surrounds his person - is there a single one who feels that breath? I know one and only one of my old friends who felt it and told me about it, without calling it by name. And another who felt it one day, as if against his will, only to forget about it the very next day (12). Because to feel this breath and to accept it, for one of my old friends as for myself, is also to accept to look at ourselves.

​ (15) Harvest and sowing

I do not think, I will no longer think of being indignant about a wind that blows, when I have clearly seen that I am not a stranger to this wind, as a fatuity in me would have liked me to believe. And even if I [◊ 36] had been a stranger to it, my indignation would have been a very paltry offering to those who are humiliated as well as to those who humiliate, and whom I have loved both.
I was no stranger to this wind, through my connivance with contempt and fear, in this world that I had chosen. It suited me to turn a blind eye to these and many other blunders, both in my professional life and in my family life. In both, I reaped what I sowed - and what others also sowed before me or with me, both my parents (and my parents' parents...) and my new friends of yesteryear. And others besides me are now reaping the rewards of the seeds that were sown, both my children (and my children's children) and one of my pupils today, who was treated with contempt by one of my pupils in the past.

And there is no bitterness or resignation in me, no self-pity, when I speak of sowing and reaping. For I have learned that even in the bitter harvest there is substantial flesh which it is up to us to nourish. When this substance is eaten and becomes part of our flesh, the bitterness disappears, which was only the sign of our resistance to a food destined for us.

And I also know that there are no harvests that are not also the sowing of other harvests, often more bitter than those that preceded them. There are still times when something in me tightens at the seemingly endless chain of carefree sowing and bitter harvesting, passed on and repeated from generation to generation. But I am no longer overwhelmed by it or revolted by it as if it were a cruel and inescapable fate, and even less am I its complacent and blind prisoner, as I once was. For I know that there is a nourishing substance in everything that happens to me, whether the seed is sown by me or by someone else - it is up to me to eat and see it transformed into knowledge. And it's no different for my children and all those I have loved and those I love at this moment, when they reap what I sowed in times of fatuity and carelessness, or what I happen to sow even today.

IV THE DOUBLE FACE

Contents

  1. Marshes and front rows
  2. Terry Mirkil
  3. Twenty years of fatuity - or the indefatigable friend
  4. The world without love
  5. A world without conflict?
  6. A well-kept secret
  7. Bourbaki, or my great luck - and its downside
  8. De Profundis
  9. My farewell - or strangers

​ (16) Marshes and front rows

But I have not yet reached the end of this reflection, on my part in the appearance of contempt and in its progression, in this world to which I blithely continued to refer by the name of ‘mathematical community’. It is this reflection, I feel now, that is the best I have to offer to those I have loved in this world, at a time when I am preparing, not certainly to return to it, but to express myself once again in it.

Above all, I think, I need to examine the kind of relationships I had with the people who were part of that world, when I was still part of it like them.

Thinking about it now, I'm struck by the fact that there was a whole part of that world that I used to rub shoulders with regularly, but which escaped my attention as if it hadn't existed. I must have perceived it at the time as a kind of ‘swamp’ with no well-defined function in my mind, not even that of a ‘sounding board’, I suppose - as a kind of grey, anonymous mass of those who invariably sat in the back rows of seminars and colloquia, as if they had been assigned there by birth, those who never opened their mouths during a talk to hazard a question, certain as they must have been in advance that their question could only be off the mark. If they asked a question of people like me, who were considered to be ‘in the know’, it was in the corridors, when it was obvious that ‘the experts’ weren't pretending to want to talk to each other - they then asked their question quickly and as if on tiptoe, ashamed of taking up the precious time of important people like us. Sometimes the question seemed to be off the mark and I tried (I imagine) to explain in a few words why; often it was relevant and I answered it as best I could, I think. In both cases it was rare for a question asked in such a mood (or, should I say, in such an atmosphere) to be followed up by a second question, which would have clarified it or expanded on it. Perhaps we, the people in the front rows, were in too much of a hurry in these cases (even though we were certainly sometimes trying not to appear so), for the fear in front of us to dissipate, and for an exchange to take place. Of course, I sensed, as did my interlocutor, what was false and artificial about the situation in which we were involved - without my ever having formulated it to myself, and without him, no doubt, ever having formulated it to himself.

The two of us functioned like strange automatons, and a strange connivance bound us together: that of pretending to ignore the anguish that embraced one of us, obscurely perceived by the other - that particle [◊ 38] of anguish in the anguish-laden air that saturated the place, which all surely perceived as we did, and which all chose to ignore by common agreement (13).

This confused perception of anguish didn't become conscious in me until the aftermath of the first ‘awakening’, in 1970, when this ‘swamp’ came out of the semi-darkness in which I had been happy to keep it in my mind until then. Suddenly, most of my new friends were precisely those who, just a year before, I would have tacitly placed in that nameless, featureless land. The so-called marsh was suddenly alive and kicking with the faces of friends linked to me by a shared adventure - another adventure!

​ (17) Terry Mirkil

To tell the truth, even before this crucial turning point, I had made friends with comrades (who later became ‘colleagues’) whom I would probably have located in the ‘swamp’, had the question been put to me (and had they not been my friends...). It took this reflection, and me digging through my memories, to remember and for the scattered memories to come together. I got to know these three friends in the very early days, when I was learning the trade in Nancy like them - at a time when we were still in the same boat, when nothing designated me as an ‘eminence’. It's probably no coincidence that there were no other such friendships in the twenty years that followed. The four of us were foreigners, and that was certainly a significant link - my relations with the young ‘normaliens’, parachuted into Nancy like me, were much less personal, and we hardly saw each other except at university. One of my three friends emigrated to South America one or two years later. Like me, he was a research associate at the CNRS, and I had the impression that he didn't really know what he was ‘looking for’. We continued to see or write to each other from time to time, and eventually we lost touch. My relationship with the other two friends lasted longer, was stronger and much less superficial. Our mathematical interests played little or no part in it.

With Terry Mirkil and his wife, Presocia, slim and fragile as he was [◊ 39], with an air of gentleness in both of them, we often spent evenings, and sometimes nights, in Nancy singing, playing the piano (Terry was playing at the time), talking about music, which was their passion, and about other important things in our lives. Not the most important things, it's true - not the things that are always so carefully hushed up... But I got a lot out of that friendship. Terry had a finesse and a discernment that I lacked, when most of my energy was already focused on mathematics. Much more than me, he had retained a sense of the simple, essential things - the sun, the rain, the earth, the wind, singing, friendship...

After Terry had found a position to his liking at Dartmouth College, not so far from Harvard, where I was a frequent visitor (from the late 1950s onwards), we continued to meet and write to each other. In the meantime, I knew that he was prone to depression, which led to long stays in ‘madhouses’, as he called them in the only terse letter he ever wrote to me, following one of those ‘horrible stays’. When we met, there was never any mention of them - except once or twice, very incidentally, to answer my astonishment that he and Presocia weren't adopting any children. I don't think it ever occurred to me that he and I could talk about the substance of the problem, or even touch on it - certainly not even the idea that there might be problems to be looked at, in my friend's life or in mine... There was an unspoken, impassable taboo about these things.

Gradually, the meetings and letters became less frequent. It's true that I was becoming more and more a prisoner of tasks and of a role, and above all of this desire, which had become like a fixed idea, perhaps an escape from something else, to constantly surpass myself in the accumulation of works - while my family life was mysteriously, inexorably deteriorating...

When I learned one day, through a letter from one of Terry's colleagues at Dartmouth, that my friend had committed suicide (it was long after he was already dead and buried...), this news came to me as if through a fog, like an echo from a very distant world that I had left behind, God knows when. A world inside me, perhaps, that had died long before Terry ended his life, devastated by the violence of an anguish that he hadn't known or wanted to resolve, and that I hadn't known or wanted to guess...

​ (18) Twenty years of fatuity - or the indefatigable friend

[◊ 40] My relationship with Terry was not distorted, at any time I believe, by the difference in our status in the mathematical world, or by any feeling of superiority that I might have derived from it. This friendship, and one or two others that life gave me in those days (without worrying whether I ‘deserved’ it!), was surely one of the rare antidotes at the time against a secret fatuity, fuelled by social status and, even more, by the awareness that I had acquired of my mathematical power and the value that I myself placed on it. The same was not true of my relationship with the third friend. Over the years, he and later his wife (whom he had met when we first met in Nancy) showed me a warm friendship, marked by delicacy and simplicity, whenever we met, in their house or mine. In this friendship there was clearly no ulterior motive, linked to status or cerebral abilities. Yet my relationship with them remained marked for more than twenty years by that deep ambiguity in me, that division I mentioned, which has marked my life as a mathematician. In their presence, each time again, I couldn't help but feel their affectionate friendship and respond to it, almost unwillingly! At the same time, for over twenty years I managed to pull off the feat of looking at my friend with disdain, from the height of my stature. It must have started in the early years in Nancy, and for a long time my prejudice extended to his wife, as if it could only be understood in advance that his wife could only be as ‘insignificant’ as he was.

Between my mother and me, we made a point of referring to him only by a mocking nickname, which must have stayed with me long after my mother's death in 1957. It now appears to me that at least one of the forces behind my attitude was the influence that my mother's strong personality exerted on me throughout her life, and for almost twenty years after her death, during which I continued to be imbued with the values that had dominated her own life. My friend's gentle, affable, non-combative nature was tacitly classed as ‘insignificant’, and became the object of mocking disdain. It is only now, taking the trouble for the first time to examine what that relationship was like, that I am discovering the full extent of the stubborn isolation from the warm sympathy of others that marked it for so long. My friend Terry, no more combative or forceful than this other friend, had [◊ 41] the good fortune to be approved of by my mother and was not the object of her mockery - and I suspect that this is why my relationship with Terry was able to flourish without inner resistance within me. His investment in mathematics was no more fervent, nor were his ‘gifts’ any more prominent, yet I never used this as an excuse to cut myself off from him and his wife with this shell of disdain and smugness!

What is still incomprehensible to me in this other relationship is that my friend's affectionate friendship was never discouraged by the reluctance he could not fail to sense in me, at each new meeting. Yet today I know that I was something other than that shell and that disdain, something other than a cerebral muscle and a conceit that took pride in it. As in them, there was the child in me - the child I tried to ignore, the object of disdain. I had cut myself off from him, and yet he lived somewhere inside me, healthy and vigorous as the day I was born. The affection of my friends, less cut off from their roots than I was, surely went to the child. And it was surely he too who responded in secret, on the sly, when the Big Chief's back was turned.

​ (19) The world without love

Fortunately, the Big Chief has aged, he's become a little more effete, and the kid has since been able to take it all in his stride. As for this relationship with these really tough friends, it seems to me that I've put my finger on the most blatant, the most grotesque case in my life of the effects of a certain fatuity (among other things) in a personal relationship. Maybe I'm fooling myself again, but I think it's also the only case where my relationship with a colleague or a friend in the mathematical world (or even elsewhere) has been invested in a lasting way by fatuity, instead of just appearing occasionally, discreetly and fleetingly. It seems to me, moreover, that among the many friends I had in the mathematical world at the time and whom I liked to keep company with, there is not one for whom I could imagine that they had experienced a similar lapse in their relationship with a colleague, whether a friend or not. Of all my friends, I was perhaps the least ‘cool’, the most ‘polite’, the least inclined to show a hint of humour (it only came to me later), the most inclined to take myself terribly seriously. In fact, I probably wouldn't have sought out the company of people like me (assuming there were any)!

[42] The amazing thing was that my friends, swamp or no swamp, put up with me and even took a liking to me. That's a good and important thing to say here - even though we often saw each other only to discuss maths for hours and days on end : affection flowed, as it still does today, between the friends of the time (according to sometimes fortuitous affinities) and me, from that first moment when I was received with affection in Nancy, in 1949, in the house of Laurent and Hélène Schwartz (where I was a bit part of the family), that of Dieudonné, that of Godement (which at one time I also visited regularly).

This affectionate warmth that surrounded my first steps in the world of mathematics, and which I have tended to forget, has been important throughout my life as a mathematician. It was certainly this warmth that gave a similar tone to my relationship with the environment that my elders embodied for me. It gave all its strength to my identification with this community, and all its meaning to the name ‘mathematical community’.

Clearly, for many young mathematicians today, it is being cut off during their apprenticeship, and often well beyond, from any current of affection or warmth; seeing their work reflected in the eyes of a distant boss and in his parsimonious comments, rather as if they were reading a circular from the Ministry of Research and Industry, which clips the wings of work and deprives it of any deeper meaning than that of a dull and uncertain livelihood.

But I am anticipating, by speaking of this disgrace, the deepest of all perhaps, of the mathematical world of the 1970s and 1980s - the mathematical world where those who were my students, and the students of my friends of yesteryear, set the tone. A world where, often, the boss assigns his subject to the student, like throwing a bone to a dog - that or nothing! Like assigning a cell to a prisoner: that's where you'll purge your solitude! Where a meticulous, solid piece of work, the fruit of years of patient effort, is rejected with the smiling contempt of the person who knows everything and has the power in his hands: ‘This work doesn't amuse me! So much for the dustbin...

Such disgraces, I am well aware, did not exist in the milieu I knew, among the friends I haunted, in the 1950s and 1960s. It is true that I learnt in 1970 that this was rather the daily bread [◊ 43] in the scientific world outside maths - and even in maths it was apparently not so rare, open contempt, flagrant abuse of power (and no recourse), even among certain renowned colleagues whom I had had the opportunity to meet. But in the circle of friends that I had naively taken to be ‘the’ mathematical world, or at least a faithful miniature expression of it, I knew nothing of the sort.

However, the seeds of contempt must already have been there, sown by my friends and by me, and which sprang up in our students. And not only in our students, but also in some of my former companions and friends. But my role is not to denounce or even to fight: you can't fight corruption. When I see it in one of my students whom I once loved, or in one of my former companions, something inside me tightens - and rather than accepting the knowledge that pain brings me, I often refuse the pain and struggle, taking refuge in refusal and a fighting attitude: such a thing has no place! And yet it does - and I even know deep down what it means. In more ways than one, I'm no stranger to it, if one of my former students or companions whom I loved likes to discreetly crush another whom I love and in whom he recognises me.

Once again I digress, doubly so I might say - as if the wind of contempt only blew around my home! Yet it is by blowing on me above all and on those who are close and dear to me that I am touched and know it. But the time is not ripe to talk about it, except to myself alone, in silence. Instead, it's time for me to pick up the thread of my reflection-testimony, which could well be called ‘In Pursuit of Contempt’ - contempt in myself and those around me, in the mathematical environment that was mine in the 1950s and 1960s.

​ (20) A world without conflict?

I'd thought I'd mention the ‘swamp’ in a few lines, just to say that it was there but that I didn't go there - and as is so often the case in meditation (and also in mathematical work), the ‘nothing’ we look at turns out to be rich in life and mystery, and in hitherto neglected knowledge. Like that other ‘nothing’, which also happened to be in Nancy (definitely the cradle of my new identity!), the ‘nothing’ of that student who was probably a bit of a loser and got treated like... I thought of it again in a flash earlier, when I wrote (perhaps a bit hastily?) that ‘these disgraces’ didn't yet exist ‘with us’. Let's just say that this is the only [◊ 44] incident of its kind that I can report, which (admittedly) resembles the ‘disgrace’ I was referring to, without dwelling too much on a detailed description. Those who have experienced it will know what I mean, without having to draw a picture. And also those who, without having experienced it, are not in a hurry to close their eyes every time they are confronted with it. As for the others, those who despise to their heart's content as well as those who are content to close their eyes (as I myself did successfully for twenty years), even an album of drawings would be a wasted effort...

It remains for me to examine my personal and professional relationships with my colleagues and pupils over these two decades, and incidentally also what I was able to learn about the relationships of my closest colleagues with each other, and with their pupils. What strikes me most today is the extent to which conflict seems to have been absent from all these relationships. I have to add straight away that this is something that in those days seemed quite natural to me - like the least of things. Conflict, between people of good will, mentally and spiritually mature and all that (the least of things, once again!), had no place. Where there was conflict, I looked on it as a sort of regrettable misunderstanding: with the right amount of goodwill and by explaining things to each other, it could only be resolved as quickly as possible and without leaving any traces! If I chose mathematics as my favourite subject from an early age, it was surely because I felt that this was the path where my vision of the world had the best chance of not coming up against disturbing denials at every step. When something has been demonstrated, after all; everyone is in agreement, that is to say, people of good will and all that, of course.

As it happens, I was right. And the story of those two decades spent in the peace and quiet of the ‘conflict-free’ (?) world of my beloved ‘mathematical community’, is also the story of a long inner stagnation inside me, eyes and ears plugged, learning nothing except maths or little else - while in my private life (first in the relationship between my mother and me, then in the family I founded immediately after her death) a silent destruction was rampant that at no time during those years did I dare to look at. But that's another story... The ‘awakening’ of 1970, about which I have often spoken in these lines, was a turning point not only in my life as a [◊ 45] mathematician, and a radical change of environment, but also a turning point (give or take a year) in my family life. It was also the year when for the first time, in contact with my new friends, I risked an occasional glance, still very furtive, at the conflict in my life. It was the moment when a doubt began to dawn on me, which matured over the years that followed, that the conflict in my life, and the conflict that I sometimes feared in the lives of others, was not just a misunderstanding, a ‘blip’ that could be wiped away with a sponge.

This (at least relative) absence of conflict, in the environment I had chosen as my own, seems to me in retrospect to be a rather remarkable thing, when I have come to learn that conflict rages everywhere humans live, in families as well as in workplaces, be they factories, laboratories or teachers‘ or assistants’ offices. It's almost as if I'd stumbled, in September or October 1948, landing unsuspectingly in Paris on the only heavenly island in the Universe where people live without conflict with each other!
All of a sudden it seemed really extraordinary, after everything I'd learnt since 1970. Surely it deserves a closer look - is it a myth, or a reality? I can see the affection that flowed between so many of my friends and me, and later between students and me, I don't have to invent it - but it almost seems as if I'm obliged to invent conflict, in this heavenly world from which conflict seems banished!

It's true, in the course of this reflection I've had the opportunity to touch on two situations of conflict, each revealing an inner attitude within me: one is the incident involving the ‘null pupil’ at Nancy, the ins and outs of which I don't know between the direct protagonists. The other is a situation of conflict within myself, a division, in my relationship with the ‘indefatigable friend’ - but this never expressed itself in the form of a conflict between people, the only form of conflict generally recognised. Remarkably, in the conventional sense of the word, the relationship between these friends and me was entirely free of conflict - there was never a cloud in it. The division was in me, not in them.

I continue the census. One of the first thoughts: the Bourbaki group! During the years when I was a more or less regular member of the group, right up to the end of the 1950s, this group embodied for me the ideal of collective work carried out with respect both for the [◊ 46] seemingly infinitesimal detail in the work itself, and for the freedom of each of its members. At no time did I sense among my friends in the Bourbaki group the shadow of a hint of constraint, either on me or on anyone else, seasoned member or guest, who came to try things out to see if things would ‘click’ between him and the group. At no time was there any hint of a struggle for influence, be it over differences of opinion on this or that issue on the agenda, or a rivalry for hegemony over the group. The group functioned without a leader, and no one apparently aspired in their heart of hearts, as far as I could see, to play that role. Of course, as in any group, one member exerted a greater influence on the group, or on other members, than another. Weil played a special role in this respect, which I have already mentioned. When he was present, he was a bit of a ‘playmaker’ (14). Twice, I think, my sensibilities got the better of me, and I left - these are the only signs of ‘conflict’ that I know of. Gradually, Serre exerted an influence on the group comparable to that of Weil. When I was a member of Bourbaki, this did not give rise to any rivalry between the two men, and I am not aware of any enmity that may have developed between them later on.

With the benefit of twenty-five years' hindsight, Bourbaki, as I knew him in the 1950s, still seems to me to be an example of remarkable success in terms of the quality of relationships within a group formed around a common project. This quality of the group seems to me to be even rarer than the quality of the books that came out of it. It was one of the many privileges of my life, full of privileges, to have met Bourbaki, and to have been part of it for a few years. If I did not stay, it was not because of conflicts or because the quality I mentioned had deteriorated, but because more personal tasks attracted me even more strongly, and I devoted all my energy to them. Moreover, this departure cast no shadow over my relationship with the group or with any of its members.

I would have to go through all the conflict situations in which I was involved, which pitted me against one of my colleagues or one of my students, between 1948 and 1970. The only thing that stands out in any way are the two brief falling-outs with Weil, which have already been mentioned. A few fleeting shadows, very fleeting shadows on my relations with Serre, because of my [◊ 47] susceptibility to a certain sometimes disconcerting casualness that he had in cutting short an interview when he had finished being interested in it, or in expressing his lack of interest, or even his aversion to such and such a work in which I was engaged, or such and such a vision of things on which I insisted, perhaps a little too much and too often! It never developed into a falling out. Apart from the differences in temperament, our mathematical affinities were particularly strong, and he must have felt, as I did, that we complemented each other.

The only other mathematician with whom I had a comparable and even stronger affinity was Deligne. On this subject, I remember that the question of Deligne's appointment to the IHES in 1969 gave rise to tensions, which I did not perceive at the time as a ‘conflict’ (which would have been expressed, say, by a falling out, or by a turning point in a relationship between colleagues).

It seems to me that I've come full circle - that at the level of conflict between people, visible through tangible manifestations, in the relationships between colleagues or between colleagues and pupils in the environment I haunted, that's all during those twenty-two years, incredible as that may seem. In other words, there was no conflict in the paradise I had chosen - so no scorn? Another contradiction in mathematics?

I'll definitely have to take a closer look!

​ (21) A well-kept secret

Yesterday I certainly forgot a few minor episodes, such as temporary ‘cold spots’ in my relationship with a colleague, due in particular to my susceptibility. I should also add three or four occasions when my self-esteem was disappointed, when colleagues and friends did not remember, in some of their publications, that an idea or result I had shared with them must have played a role in their work (so it seemed to me). The fact that I still remember it shows that it was a sensitive point, and one that perhaps hasn't entirely disappeared with age! Except on one occasion, I refrained from mentioning it to those concerned, whose good faith was certainly above suspicion. The opposite situation must also have occurred, but I never heard anything about it. I am not aware of any case, in my ‘microcosm’, where a question of priority was the occasion of a [◊ 48] quarrel or enmity, or even of bitter-sweet remarks between the parties concerned. Still, the one time I had such a discussion (in what seemed to me to be an egregious case), there was a spat of sorts, which cleared the atmosphere without leaving a residue of resentment. This was a particularly brilliant colleague who had, among other abilities, the ability to assimilate with impressive speed everything he heard, and it seems to me that he often had an unfortunate tendency to take as his own the ideas of others that he had just learned from their mouths.

There is a difficulty here which must be found in a more or less strong form in all mathematicians (and not only in them), and which is not only due to the egotistical drive which pushes most of us (and I am no exception) to attribute ‘merits’ to ourselves, whether real or supposed. The understanding of a situation (mathematical or otherwise), however we achieve it, with or without the assistance of others, is in itself something of a personal essence, a personal experience whose fruit is a vision, also necessarily personal. A vision can sometimes be communicated, but the vision communicated is different from the initial vision. That being the case, we need to be very vigilant in determining the part played by others in the formation of our vision. I'm sure I didn't always have this vigilance, which was the least of my worries, even though I expected it from others! Mike Artin was the first and only person to suggest to me one day, with the joking air of someone divulging an open secret, that it was both impossible and perfectly pointless to bother trying to discern which part was ‘one's own’ and which was ‘someone else's’ when you manage to take a substance in hand and understand something about it. It was a bit disconcerting, even though it was not at all part of the deontology that had been taught to me by Cartan, Dieudonné, Schwartz and others. Yet I had a vague feeling that there was a truth in his words, and just as much in his laughing gaze, that had eluded me until then\(^1\) . My relationship with mathematics (and above all, with mathematical production) was heavily invested by the ego, and this was not the case with Mike. He really gave the impression of doing maths like a kid having fun, without forgetting to eat and drink.

​ (22) Bourbaki, or my great luck - and its downside

Even before plunging a little further beneath the visible surface, there is one observation that [◊ 49] imposes itself on me right now: it is that the mathematical milieu I inhabited for two decades, in the 1950s and 1960s, was indeed a ‘world without conflict’, so to speak! That's quite an extraordinary thing in itself, and it deserves a bit of thought.

I should point out straight away that this was a very restricted milieu, the central part of my mathematical microcosm, limited to my immediate ‘environment’ - the twenty or so colleagues and friends I met regularly, and with whom I had the strongest ties. Reviewing them, I was struck by the fact that more than half of these colleagues were active members of Bourbaki. It is clear that the core and soul of this microcosm was Bourbaki. It was, more or less, Bourbaki and the mathematicians closest to Bourbaki. In the 1960s I was no longer part of the group myself, but my relationship with some of the members remained as close as ever, particularly with Dieudonné, Serre, Tate, Lang and Cartier. I continued to be a regular at the Bourbaki seminar, or rather I became one at that time, and it was at that time that I gave most of my talks there (on the theory of schemas).

It was undoubtedly in the 1960s that the ‘tone’ in the Bourbaki group shifted towards an increasingly pronounced elitism, of which I was certainly a part at the time, and which for that reason I was unlikely to notice. I still remember how astonished I was, in 1970, to discover the extent to which the very name of Bourbaki had become unpopular with large sections of the mathematical world (which I had hitherto ignored), as being more or less synonymous with elitism, narrow dogmatism, the cult of the ‘canonical’ form at the expense of a living understanding, hermeticism, castrating anti-spontaneity, and so on! It wasn't just in the ‘swamp’ that Bourbaki got a bad press: in the 1960s, and perhaps even earlier, I had heard occasional echoes of it from mathematicians with a different mindset, allergic to the ‘Bourbaki style’ (15). As an unconditional adherent, I was surprised and a little saddened - I thought that mathematics brought people together! And yet I should have remembered that when I started out, it wasn't always easy or inspiring to swallow a Bourbaki text [◊ 50], even if it was expeditious. The canonical text hardly gave an idea of the atmosphere in which it was written, to say the least. It now seems to me that this is precisely the main shortcoming of the Bourbaki texts - that not even an occasional smile can give rise to the suspicion that these texts were written by people, and people bound by something other than some oath of unconditional fidelity to ruthless canons of rigour....

But the question of the slide towards elitism, like that of Bourbaki's writing style, is a digression. What strikes me here is that, the ‘Bourbakian microcosm’ that I had chosen as my professional environment, was a world without conflict. This seems to me all the more remarkable given that the protagonists in this milieu each had a strong mathematical personality, and many are considered to be ‘great mathematicians’, each of whom certainly had the weight to form his own microcosm, of which he would have been the centre and undisputed leader! (16) It is the cordial and even affectionate cohabitation, for two decades, of these strong personalities in the same microcosm and in the same working group, that strikes me as so remarkable, perhaps unique. This ties in with the impression of ‘exceptional success’ that was already expressed yesterday about Bourbaki.

In the end, it would seem that I was exceptionally lucky, when I first came into contact with the world of mathematics, to have stumbled upon the privileged place, in time and space, where a mathematical community of exceptional quality, perhaps unique for that quality, had been forming for some years. This environment became mine, and has remained for me the embodiment of an ideal ‘mathematical community’, which probably did not exist at that time (beyond the environment that for me embodied it) any more than at any other time in the history of mathematics, except perhaps in a few equally restricted groups (such as perhaps the one that had formed around Pythagoras in a quite different spirit).

My identification with this milieu was very strong, and inseparable from my new identity as a mathematician, born at the end of the 1940s. It was the first group, beyond the family group, where I was warmly welcomed and accepted as one of them. Another link, of a different kind: my own [◊ 51] approach to mathematics found confirmation in that of the group, and in that of the members of my new environment. It was not identical to the ‘Bourbachian’ approach, but it was clear that the two were sisters.

This environment, moreover, must have represented for me that ideal place (or very close to it!), that place without conflict whose quest had undoubtedly led me to mathematics, the science of all sciences where any hint of conflict seemed to me to be absent! And although I spoke earlier of my ‘exceptional good fortune’, it was clear to me that this good fortune had its downside. While it enabled me to develop my skills and to show my worth as a mathematician in the midst of my elders, who have become my peers, it was also a welcome escape from conflict in my own life, and a long period of spiritual stagnation.

​ (23) De Profundis

This ‘Bourbach’ environment has certainly had a strong influence on me as a person and on my vision of the world and my place in it. This is not the place to try and pinpoint this influence and how it has played out in my life. I would only say that it does not seem to me that my inclinations towards fatuity, and their meritocratising rationalisations, were in any way stimulated by my contact with Bourbaki and my insertion into the ‘Bourbachean milieu’ - at least not in the late 1940s and 1950s. The seeds had been sown in me for a long time, and would have found an opportunity to develop in any other environment. The incident of the ‘null student’ that I have reported is in no way typical, quite the contrary, of an atmosphere that would have prevailed in that environment, I repeat, but only of an ambiguous attitude in my own person. The atmosphere in Bourbaki was one of respect for the individual, an atmosphere of freedom - at least that's how I felt; and it was such as to discourage and attenuate any inclination towards attitudes of domination or fatuity, whether individual or collective.

This environment of exceptional quality is no more. I don't know when it died, but no one, no doubt, realised it and sounded the death knell, even in their own hearts. I suppose that there must have been an insensitive deterioration in people - we all had to ‘grow a pair’, become stale. We became important people, listened to, powerful, feared, sought after. [52] Perhaps the spark was still there, but the innocence was lost along the way. Some of us may find it again before we die, like a new birth - but this environment that welcomed me is no more, and it would be pointless for me to expect it to resurrect. Everything is back to normal.

And perhaps respect has also been lost along the way. By the time we had pupils, it was perhaps too late for the best to be passed on - there was still a spark, but no more innocence or respect, except for ‘his peers’ and ‘his own’.

The wind can rise and blow and burn - we are sheltered behind thick walls, each of us with ‘our own’.

Everything is back to normal...

​ (24) My farewell - or strangers

This retrospective of my life as a mathematician takes a completely different path than I had planned. To tell the truth, I wasn't even thinking of a retrospective, but only of saying in a few lines, or even a page or two, what my relationship was today with this world that I had left, and perhaps also, conversely, what was the relationship to me of my former friends, according to the echoes that reach me from far and wide. I had intended, on the other hand, to take a closer look at the sometimes strange vicissitudes of some of the ideas and notions that I had introduced during these years of intense mathematical work - I should say rather: the new types of objects and structures that I had the privilege of glimpsing and drawing out of the night of the totally unknown into the twilight, and sometimes even into the clearest light of day! Now, in what has become a meditation on the past, in an effort to better understand and come to terms with a certain, sometimes bewildering, present, this statement seems to stand out. Decidedly, the planned reflection on a certain ‘school’ of geometry, which was formed at my instigation, and which vanished without (almost) leaving any trace, will have to wait for a more propitious occasion\(^2\) . For the time being, therefore, my concern will be to bring to a conclusion this retrospective on my life as a mathematician in the world of mathematicians, not to epilogise on a work and its fate.

During the last five days, when I have been busy with tasks other than these reflective notes, one memory has come back to me with a certain [◊ 53] insistence. It will serve as an epilogue to the De Profundis on which I had stopped.

It takes place towards the end of 1977. A few weeks earlier, I had been summoned to appear before the Montpellier criminal court for the offence of having ‘gratuitously housed and fed a foreigner in an irregular situation’ (that is, a foreigner whose residence papers in France were not in order). It was at the time of this quotation that I learned of the existence of this incredible paragraph of the 1945 ordinance governing the status of foreigners in France, a paragraph that prohibits any French person from providing assistance in any form whatsoever to a foreigner ‘in an irregular situation’. This law, which had no analogue even in Hitler's Germany with regard to the Jews, had apparently never been applied in its literal sense. By a very strange ‘coincidence’, I had the honour of being taken as the first guinea pig for the first application of this unique paragraph.

For a few days I was stunned, paralysed and deeply discouraged. Suddenly I felt like I'd gone back thirty-five years, to a time when life didn't weigh much, especially for foreigners... Then I reacted, I shook myself out of it. For a few months I put all my energy into trying to mobilise public opinion, first in my university and in Montpellier, and then at national level. It was during this period of intense activity, for a cause that later proved to be lost in advance, that the episode that I could now call my farewell took place.

With a view to taking action at national level, I had written to five well-known ‘personalities’ in the scientific world (including a mathematician) to inform them of this law, which even today still seems as incredible as the day I was quoted. In my letter, I proposed a joint action to demonstrate our opposition to this scurrilous law, which was tantamount to outlawing hundreds of thousands of foreigners living in France, and singling out for public suspicion, like lepers, millions of other foreigners, who at once became suspects, likely to get the French into the worst trouble if they weren't on their guard.

[54] Surprisingly, and completely unexpectedly for me, I didn't receive a reply from any of these five ‘personalities’. Decidedly, I had things to learn...

It was then that I decided to go to Paris for the Bourbaki seminar, where I was sure to meet up with many old friends, in order first of all to mobilise opinion in the mathematical community, with which I was most familiar. This community, it seemed to me, would be particularly sensitive to the cause of foreigners, since all my mathematical colleagues, like myself, have to deal on a daily basis with foreign colleagues, pupils and students, most if not all of whom have had problems with their residency papers, and have had to face arbitrariness and often contempt in the corridors and offices of police headquarters. Laurent Schwartz, whom I had told about my project, told me that I would be given the floor at the end of the presentations on the first day of the seminar to explain the situation to the colleagues present.

So that day I arrived with a bulky packet of leaflets in my suitcase for my colleagues. Alain Lascoux helped me distribute them in the corridor of the Institut Henri-Poincaré, before the first session and during the ‘intermission’ between the two presentations. If I remember correctly, he even produced a small leaflet of his own - he was one of the two or three colleagues who, having heard about the affair, were moved and contacted me before my trip to Paris to offer their help (17). Roger Godement was one of them, and he even produced a leaflet with the headline ‘A Nobel Prize winner in prison’. It was chic of him, but we were definitely not on the same wavelength: as if the scandal was to attack a ‘Nobel Prize winner’, rather than the first lampoonist who came along!
There were a lot of people there on the first day of the Bourbaki seminar, and a lot of people I had known more or less closely, including Bourbaki's friends and companions from the old days; I think most of them were there. Several of my former students too. It must have been ten years since I last saw all these people, and I was glad to have this opportunity to see them again, even if it meant seeing a lot of them at once! But we'd end up meeting up again in smaller numbers...

The reunion, however, ‘wasn't that’, and that was quite clear from the very [◊ 55] beginning. Many hands extended and shaken, for sure, and many questions ‘here, you here, what wind brings you?’, yes - but there was like an air of indefinable awkwardness behind the cheerful tones. Was it because they weren't really interested in the cause that had brought me here, even though they had come for some tri-annual mathematical ceremony that demanded all their attention? Or, irrespective of what brought me here, was it my very person who inspired this discomfort, rather like the discomfort that a defrocked priest would inspire among a group of good-natured seminarians? I couldn't say - perhaps it was both. For my part, I couldn't help noticing the transformation that had taken place in certain faces that had once been familiar, even friendly. They had frozen, you might say, or slumped. A mobility that I had known there seemed to have disappeared, as if it had never been. I found myself standing before strangers, as if nothing had ever linked me to them. Somehow I sensed that we weren't living in the same world. I'd thought I'd found my brothers on this exceptional occasion, and here I was before strangers. Admittedly well behaved, I don't remember any bitter-sweet comments or leaflets lying around. In fact, all (or almost all) of the leaflets handed out must have been read, curiosity permitting.

But that doesn't mean that the scurrilous law has been jeopardised! I had my five minutes, perhaps even ten, to talk about the situation of those whom I considered to be brothers, called ‘foreigners’. There was a packed amphitheatre of colleagues there, quieter than if I'd been giving a mathematical lecture. Perhaps there was no longer the conviction to speak to them. There was no longer, as there used to be, a current of sympathy and interest. There must have been a lot of people in a hurry, I thought, so I cut the meeting short, proposing that we meet again immediately, with the colleagues who felt concerned, to discuss in more detail what could be done...

When the meeting was declared adjourned, there was a general rush to the exits - obviously, everyone had a train or metro about to leave, which they had to make sure they didn't miss! In the space of a minute or two, the Hermite amphitheatre was empty - it was like a miracle! The three of us found ourselves in the big deserted amphitheatre, under the harsh [◊ 56] lights. Three of us, including Alain and me. I didn't know the third one, one of those unmentionable foreigners again I bet, in dubious company and illegal to boot! We didn't take the time to dwell at length on the quite eloquent scene that had just unfolded before us. Perhaps I was the only one who couldn't believe my eyes, and my two friends were kind enough to refrain from commenting on the matter. I'd obviously just arrived...

The evening ended with Alain and his ex-wife Jacqueline taking stock of the situation and reviewing what could be done; getting to know each other a little better, too. Neither on that day, nor later, did I take the time to situate the episode I had just experienced in terms of the past. It was on that day, however, that I had to understand without words that a certain environment, a certain world that I had known and loved was no more, that a living warmth that I had thought I would find again had dissipated, probably a long time ago.

That hasn't stopped the echoes that still reach me, year after year, from that world whose warmth has fled, from disconcerting me and touching me painfully. I doubt that this reflection will change anything for the future - except, perhaps, that I will rebel less at being touched in this way...

  1. (30 September) For another aspect of things, however, see the note of 1​ June (three months later than the present text), ’Ambiguity‘ (no. 63‘’), examining the pitfalls of a certain complacency to oneself and to others.
  2. This ’more propitious opportunity‘ appeared earlier than expected, and the reflection in question is the subject of the second part, ’The Burial‘, of Harvests and Sowing.

V TEACHER AND PUPILS

Contents

  1. The student and the programme
  2. Rigour and rigour
  3. The blunder - or twenty years later
  4. The unfinished harvest
  5. The enemy father (1)
  6. The enemy father (2)
  7. The power to discourage
  8. The mathematician's ethic

​ (25) The student and the programme

I have not finished reviewing my relationships with other mathematicians, at a time when I felt that I was part of the same world, of the same ‘mathematical community’. Above all, I need to look at my relationships with my students, as I experienced them, and with others for whom I was the elder statesman.

Generally speaking, I think I can say, without reservation, that my relationships with my students were ones of respect. In this respect at least, I believe that what I had received from my elders when I was a student myself did not deteriorate over the years. As I had a reputation for doing ‘difficult’ maths (admittedly a very subjective notion!), and also for being more demanding than other bosses (already a less subjective thing), the students who came to me were highly motivated from the start: ‘they wanted it’! There was just one student who at first was a bit ‘ollé ollé’, it wasn't really clear whether he was going to start - and then he did, he started without me having to push...

[57] As far as I can remember, I accepted all the students who asked to work with me. For two of them, it turned out after a few weeks or months that my style of working didn't suit them. To tell the truth, it seems to me now that on both occasions these were blocking situations, which I hastily interpreted as signs of an inability to work mathematically. Today I would be much more cautious about making such predictions. I had no hesitation in sharing my impressions with the two people concerned, advising them not to continue in a career which, it seemed to me, was not suited to them. In fact, I knew that for one of these two students at least, I had made a mistake - this young researcher went on to make a name for himself in difficult subjects at the frontiers of algebraic geometry and number theory. I don't know whether the other student, a young woman, continued or not after her disappointment with me. It cannot be ruled out that my impression of her abilities, expressed too peremptorily, discouraged her, even though she was perhaps just as capable as anyone else of doing a good job. It seems to me that I had given credit and confidence to these students as to the others. On the other hand, I lacked the discernment to distinguish between what were surely signs of blocking, rather than ineptitude (18).

From the early 1960s onwards, over a period of ten years, eleven students worked with me on their doctoral theses (19). Having chosen a subject to suit them, they each did their work with gusto, and (as I felt) they identified strongly with the subject they had chosen. There was, however, one exception, in the case of a student who had chosen, perhaps without any real conviction, a subject that ‘had to be done’, but which also had some thankless aspects, since it involved a technical fine-tuning, sometimes arduous, even dry, of ideas that had already been acquired, at a time when there were hardly any surprises or suspense in prospect (20). Carried away by the needs of a vast programme for which I needed manpower, I must have lacked psychological discernment in proposing this subject, which was certainly not suited to the particular personality of this student. I'm sure he didn't realise what he was getting himself into! In any case, neither he nor I were able to see in time that things had got off on the wrong foot and that it would be better to start again on something else.

[◊ 58] Clearly he was working without any real conviction, and always looking a bit sad and sullen. I think I had already reached a point where I didn't pay too much attention to these things, which nevertheless (I should have remembered) are the day and night of all research work, and not just research! My role then was limited to being annoyed when the work seemed to be dragging on, and breathing a sigh of relief when it got going again, then when the planned programme was finally ‘completed’.

It was only years after my awakening in 1970, having corresponded with this former pupil (now a teacher, like everyone else in these merciful times!), that the idea came to me that something had definitely gone wrong in this case, that it was perhaps not a total success. Today, I see it as a failure, despite the ‘completed programme’ (by no means a botched one!), the diploma and the job. And I bear a large part of the responsibility for having put the needs of a programme ahead of those of a person - a person who had trusted me. The ‘respect’ I claimed to have shown my students (‘without any reservations’) was superficial, divorced from the very essence of respect: loving attention to the person's needs, at least insofar as their satisfaction depended on me. The need, here, for joy in the work, without which it loses its meaning and becomes a constraint.

In the course of this reflection, I had occasion to speak of a ‘world without love’, and I was looking for the seeds of that world in myself, which I rejected. Well, here's a big one - and I can't say today how it sprang up in others. This superficial respect, devoid of attention and true love, is the ‘respect’ I also gave my children. With them, I had the privilege of seeing this seed grow and proliferate. And I've come to understand that there's no point in begrudging the harvest...

​ (26) Rigour and rigour

With the exception of this one pupil, who was certainly no less ‘gifted’ than the others, I can say that relations between my pupils and myself were cordial, often even affectionate. By force of circumstance, they all learned to be patient with my two main faults as a ‘boss’: that [◊ 59] of having impossible handwriting (although I think all of them eventually learned to decipher me) and, something more serious of course (and which I didn't realise until much later), my fundamental difficulty in following other people's thoughts without first translating them into my own images and rethinking them in my own style. I was much more inclined to communicate to my students a certain vision of things that I had imbibed strongly, rather than encouraging them to develop a personal vision, perhaps quite different from my own. This difficulty in relating to my students hasn't disappeared yet, but it seems to me that its effects have been attenuated by the fact that I'm aware of this tendency within myself. Perhaps my temperament, innate or acquired, predisposes me more to solitary work, which was mine for the first fifteen years of my mathematical activity (from 1945 to about 1960), than to the role of ‘teacher’ in contact with pupils whose mathematical vocation and personality are not fully formed (21). It is also true, however, that since early childhood I have loved teaching, and that from the 1960s to the present day, the students I have had have played an important role in my life. In other words, my teaching activity and my role as a teacher have played and continue to play a major role in my life (22).

During this first period of my teaching career, there was no apparent conflict between any of my students and myself, which would have been expressed even by a temporary ‘coldness’ in our relations. Only once did I find myself obliged to tell a pupil that he was not taking his work seriously and that I was not interested in continuing with him if it went on like that. Of course, he knew just as well as I did what he was talking about, he pulled himself together and the incident was closed without a trace. On another occasion, back in the early 1970s, when I was devoting most of my energy to the activities of the ‘Survivre et vivre’ group, a student to whom I had shown (as was my custom) the thesis report I had just written on his work, became angry, judging that certain considerations in the report called into question the quality of his work (which was in no way my intention). This time it was I who rectified the situation without any difficulty. It didn't seem to me at the time that this short incident could cast a shadow over our relationship, but I may have been wrong. The relationship between this pupil and myself had been more impersonal than with the [◊ 60] other pupils (apart from the ‘sad pupil’ I mentioned), a good working relationship without more, without any real warmth passing between us. I don't think, however, that it was an unconscious lack of benevolence in me that would have made me include in my report the considerations that he considered disadvantageous towards him, adding ‘that he wasn't going to let the thing go’ as a fellow student of his, who had already done his thesis with me, had done. With this other student, who was naturally sensitive and affectionate, I had a particularly friendly relationship; if I had included in my report on his thesis the same kind of consideration that had so displeased his classmate, it was surely not for lack of benevolence! Moreover, for both of them, as for all my students, I would not have given the green light for a defence if I had not been fully satisfied with the work they presented. In fact, none of my students from this period had any difficulty in finding a suitable job quickly after their thesis.

Until 1970, I had practically unlimited availability to my students (22'). When the time was ripe, and whenever it might be useful, I would spend whole days with one or other of them, if necessary, working on questions that had not been worked out, or reviewing together the successive stages of their work. As I experienced these work sessions, it doesn't seem to me that I ever played the role of ‘director’ making decisions, but that each time it was a joint research project, where discussions took place on an equal footing, until both were completely satisfied. The student contributed a considerable amount of energy, which of course was in no way comparable to what I had to contribute myself, although I had more experience and sometimes a more acute sense of smell.

However, the thing that seems to me to be the most essential for the quality of any research, whether intellectual or otherwise, is not at all a question of experience. It's the demands you make on yourself. It is not a matter of scrupulous compliance with any standards, rigorous or otherwise. It consists of an extreme attention to something delicate within ourselves, which escapes all norms and all measures. This delicate thing is the absence or presence of an understanding of the thing being examined. More precisely, the attention I want to speak of is an attention to the quality of understanding present at every moment, from the cacophony of a heterogeneous pile of notions and statements (hypothetical or known), to the total satisfaction, the complete harmony of a perfect understanding. The depth of a search, whether its outcome is a fragmentary or total understanding, lies in the quality of this attention. Such attention does not appear to be the result of a precept that one would follow, of a deliberate intention to "be careful", to be attentive — it arises spontaneously, it seems to me, from the passion to know, it is one of the signs that distinguish the drive for knowledge from its egoic counterfeits. This attention is also sometimes referred to as ’rigor.‘It is an inner rigor, independent of the canons of rigor that may prevail at a given moment in a (let's) given discipline. If in this book I allow myself to take liberties with canons of rigor (which I have taught and which have their reason for being and their usefulness), I do not believe that this more essential rigor is less than in my past publications, in canonical style. And if I have been able, perhaps, in spite of everything, to transmit to my students something of greater value than a language and a know-how, it is undoubtedly this requirement, this attention, this rigor — if not in the relationship to others and to oneself (whereas at this level it was as lacking in me as in anyone else), at least in mathematical work (23). This is, of course, a very modest thing, but perhaps, in spite of everything, better than nothing.

​ (27) The blunder - or twenty years later

Except perhaps in the case of the two students I mentioned, with whom a working relationship was not established in the end, I don't remember any of the other students who came to me to ask to work with me coming with any ‘stage fright’ or fear. No doubt they already knew me to a greater or lesser extent, having attended my seminar at the IHES for some time. If there was any awkwardness at the start of our relationship, it eventually dissipated, leaving no trace, in the course of the work. I should, however, make two exceptions here. One concerns a pupil who never really got to grips with his work, and who remained monosyllabic even during our work together. Perhaps he also came at a time when I was becoming less available, and there were no sessions of piecework with him, lasting afternoons and [◊ 62] whole days. No, in fact I don't remember any such sessions; I rather think that we mostly saw each other at the drop of a hat, for an hour or two, to take stock of where he was at. He must have had the worst time with me!

The other student I wanted to talk about, on the other hand, worked with me at a time when I was still completely available to my students. Our relationship was cordial from the start. He's even one of the few students with whom I've established a friendly relationship, the ones I used to see in their homes just as they came to mine, a sort of family-to-family relationship. It's true that even in these cases, the relationship always remained on a relatively superficial level, at least as far as I was concerned. On a conscious level, while I was already unaware of much of what was going on at home, under my own roof, I knew almost nothing about the lives of my mathematician friends, whether students or not, apart from the names of their wives and children (and even then, I sometimes forgot them, without ever being blamed!). Perhaps I was an extreme case of a ‘coward’, but I think that in the mathematical environment I knew, most if not all relationships, even friendly and affectionate ones, remained at that superficial level where you know very little about each other, apart from what is perceived at an informal level. This is surely one of the reasons why conflict between people was so rare in this environment, whereas it's clear to me that division existed within most of my colleagues and friends, and within their families, just as much as it did in my own home and everywhere else.

I don't think that my relationship with this pupil was any different from my relationship with others, nor did I feel at the time that, conversely, his relationship with me was in any significant way different from that of other pupils, particularly those with whom friendships were formed. It's only recently that I've been able to realise that this must have been a stronger relationship than for most of my other students. The visible manifestations of an unspoken conflict came as an unexpected revelation, almost twenty years after he had been my pupil. Only then did I make the connection with a long-forgotten ‘little’ fact. For a long time, perhaps even during the entire period (a few years) when we worked together more or less regularly, [◊ 63] this student had retained a certain amount of ‘stage fright’. This manifested itself at every meeting, through unmistakable signs. These signs disappeared fairly quickly afterwards, in the course of working together. Of course, I was embarrassed by these signs of unease, and I sensed that he was even more so. We both just pretended to ignore it. Surely the idea of talking about it wouldn't have occurred to either of us, nor even the idea of paying any attention to a strange situation, obviously worthy of interest! For him as for me, this ‘stage fright’ must have felt like a simple ‘burr’, which had no reason to be. We were regularly reminded of the ‘burr’, but each time it had the good taste to disappear, leaving us free to get on with serious things, like maths - and at the same time to forget ‘what didn't belong’. I don't remember stopping once to ask myself any questions about the meaning of the blunder, and I'm sure that my pupil and friend felt the same way. No doubt nothing in what we had both known around us since our early childhood could have suggested to him or to me the idea of any other attitude towards a disturbing thing than that of removing it as far as possible so that it ceased to disturb. In this case it was perfectly possible, easy even, and we were in perfect agreement that we had seen nothing, felt nothing and heard nothing.

Through many echoes and cross-checks that have come back to me over the last two or three years, I realise, however, that what we had dismissed as having no reason to be there, did not have to cease to be there, and to manifest itself. The things that sometimes come back to me don't ‘belong’ either - and yet ‘they do’, and now they can't be dismissed out of hand...

​ (28) The unfinished harvest

Until the first ‘awakening’ in 1970, my relationships with my pupils, like my relationship with my own work, were a source of satisfaction and joy, one of the tangible, indisputable foundations of a sense of harmony in my life, which continued to give it meaning, while elusive destruction raged in my family life. At that time, I saw no apparent element of conflict in these relationships, none of which was then, at any time, however fleetingly, the cause of frustration or grief. It may seem paradoxical that the conflict in the relationship [◊ 64] with one of my pupils only became apparent after this famous awakening, after a turning point which gave my life an openness it had not known before, and my person a little beginning of flexibility perhaps - qualities which, one might think, should be such as to resolve or avoid the conflict, and not to provoke or exacerbate it.

On closer examination, however, I can see that the paradox is only apparent, and that it disappears, whichever way you look at it. The first thing that comes to mind is that for a conflict to have a chance of being resolved, it must first have manifested itself. The stage of manifested conflict represents a maturing compared to that of hidden or ignored conflict, whose manifestations do exist, and are all the more ‘effective’ because the conflict expressed by them remains ignored. Also: for a conflict to manifest itself in a recognisable way, a distance must first have been reduced or disappeared. The changes that have taken place in my life over the last fifteen years, particularly in the course of successive ‘awakenings’, have all been changes, it seems to me, that have reduced a distance, erased an isolation. A conflict that finds it difficult to express itself in relation to a prestigious, admired boss, feels more at ease in relation to someone who has been stripped of a position of power (voluntarily, in this case), who has been exiled from a certain milieu holding authority and prestige, who is perceived less and less as an incarnation or a privileged representative of some entity (such as mathematics), and more and more as a person like any other : a person who is not only susceptible to harm, but who, moreover, is less and less inclined to hide from injury or pain. And thirdly and most importantly: my development since the first awakening, especially at that time and in the years that followed, was likely to raise (or perhaps awaken) questions, concern and ‘questioning’ in the well-ordered world of my former students. I had ample opportunity to realise that this was the case not only for them, but also among my friends and companions in the mathematical world of yesteryear, and sometimes even among scientific colleagues who knew me only by hearsay.

It must also be said that resolving a conflict of any depth is one of the rarest of things. More often than not, notwithstanding all the truces and surface reconciliations, the growing procession of our conflicts follows us without [◊ 65] hardly leaving us for a second throughout our lives, only to leave us in the sullen hands of the undertaker. Occasionally I have seen a conflict unravelled just a little, and sometimes even resolved in knowledge - but so far such a thing has not happened in the course and on the occasion of my relationship with one of my pupils, or with one of my erstwhile friends in the mathematical world. And I also know that it is by no means certain that such a thing will ever happen, even if I were to live another hundred years.

It is remarkable that the very moment of my break with a certain past - by which I mean the episode of my departure from the IHES (from the institution that represented something like the ‘matrix’ of the mathematical microcosm that had formed around me) - that this decisive episode was at the same time the first occasion on which one of my students expressed profound antagonism towards me. It was surely this circumstance that made this episode particularly painful, like a birth that had taken place under particularly difficult conditions. Of course, at the time I couldn't see this episode, the meaning of which escaped me, in the light in which I have since learned to see it. This painful surprise remained with me for a long time to come. And yet, in the summer of that same year, that bitter departure felt like a liberation - like a door that had suddenly opened wide (all I had to do was push it!) onto an unsuspected world, beckoning me to discover it. And each new awakening since then has also been a new liberation: the discovery of a subjection, an inner hindrance, and the rediscovery of the presence of an immense unknown, hidden behind the familiar appearance of what was supposed to be ‘known’. But throughout these fifteen years, and right up to the present day, this stubborn, discreet and unwavering antagonism has followed me, as the only major lasting source of frustration I have experienced in my life as a mathematician (23'). I could perhaps say that it was the price I paid for that first liberation, and for those that followed. But I am well aware that liberation and inner maturation have nothing to do with a ‘price to pay’, that they are not a question of ‘profits’ and ‘losses’. Or to put it another way: when the harvest is completed, when it is finished, there is no loss - the very thing that seemed like ‘loss’ has become ‘profit’. And it is becoming clear that I have not yet been able to bring this harvest to its [◊ 66] conclusion, and it remains, even as I write these lines, unfinished.

​ (29) The enemy father (1)

The kind of students who started working with me after the turning point in 1970, in the completely different environment of a provincial university, was also very different from the students before. Only two of them went on to work with me on a doctoral thesis. The work of the others was at DEA or postgraduate doctoral level. I should also include a good number of students who have taken to certain introductory research courses, which have given them the opportunity to ask themselves mathematical questions that were often unforeseen, and sometimes to devise original methods for solving them. I found the most active participation in certain ‘option courses’ for first-year students. On the other hand, for students who have already been exposed to the university environment for a few years, a certain freshness, a capacity for interest and personal vision are already more or less extinguished. Many of the students in the optional courses clearly had the makings of an excellent mathematician. Given the current situation, I was careful not to encourage any of them to go down that path, even though it could have attracted them and where they could have excelled.

With the students who took one of my ‘courses’ to prepare for a master's degree, the relationship usually didn't last beyond the year. On each occasion, I had the impression that they quickly became cordial and relaxed, on the whole. With the exception of one student who suffered from overwhelming ‘stage fright’ (23"), the same was true of students who were officially expected to prepare a research paper under my guidance, at one level or another. One difference (among many!) with my previous students was that our relationship was not so much limited to joint mathematical work. Often the exchange between the student and me involved our personalities in a less superficial way (\(23_V\) ). So it's not surprising that in this second period of my teaching career, the conflictual elements in the relationship with certain pupils came out more clearly and directly, even vehemently. Among my ex-students from the first period, there were two who subsequently displayed attitudes of systematic and [◊ 67] unequivocal antagonism (which I have had occasion to mention in passing), which nevertheless remained at the level of the informal, and perhaps even the unconscious. In the second, longer period, there were three students with whom I was confronted by antagonism. In two of them it manifested itself acutely.

With one of these students, the antagonism appeared overnight in a relationship that had been most friendly, many years after this friend had ceased to be my student. I suspect that the cause of the conflict was not so much my unspeakable conduct and personality, as a long-suppressed dissatisfaction at not having found the reception for his work (which had been excellent) that he would have been entitled to expect. This was the downside of the dubious privilege of having had me as his boss ‘after 1970’, and he must have resented me for it, without really admitting it even to himself.

With the other student, an acute antagonism had already emerged after a year and a half's work, in an atmosphere that had seemed very cordial. This is the first and only time that a relational difficulty between a student and myself has arisen at a time when he was still a student. It made it impossible for us to continue working together, even though we had started off on the right foot, with the most auspicious enthusiasm for a magnificent subject, it has to be said. I had the feeling that this young researcher had an insidious lack of confidence in his ability to do a good job (an ability that I had no doubt about), and that the manifestation of the antagonism was a sort of ‘headlong rush’ to pre-empt a dreaded failure and blame it on an odious boss (23''').

One aspect common to all these instances of conflict between students and myself, in the nearly twenty-five years that I have been teaching mathematics, is a strong ambivalence. In all these cases, without exception, the antagonism manifests itself after the fact, often insidiously, in a relationship of sympathy that can be left in no doubt. I can even say that in all these cases, as in many others where a frankly antagonistic component has not manifested itself, my person has exerted and still exerts a strong attraction. It is surely the very strength of this attraction that also feeds the strength of the antagonism and ensures its continuity. This is surely also the case in instances where antagonism takes the form of violent antipathy [◊ 68], of outraged rejection; as well as in such other cases, at the opposite extreme, where under the rigorous flag of friendly respect is expressed (when the occasion is right) an affectation of casual and delicately measured disdain....

Such situations of ambivalence, to be honest, are not peculiar to my relationship with some of my students or ex-students. In fact, they have abounded throughout my adult life, since at least the age of thirty (i.e. since my mother died). This has been true both in my love life and in my relationships with men, and more specifically with men who are much younger than I am. I've come to understand that there's something in me, innate or acquired I'm not sure how to describe it, that seems to predispose me to play the father figure. You'd think I'd have the ideal build and the right vibe to make the perfect adoptive father! It has to be said that the role of Father fits me like a glove - as if it had been mine from birth. I won't try to count the number of times I've taken on such a role vis-à-vis another person, with perfect tacit agreement on both sides. Most often this distribution of father-son or father-daughter roles remained unspoken, even unconscious, but it also happened to be formulated more or less clearly. In some cases, I acted as a father without even having played the game, I think, in ignorance of what was going on, both consciously and unconsciously.

I first became aware of the role of adopted father in 1972, at the time of ‘Survivre et vivre’, when I was suddenly confronted with an attitude of violent rejection on the part of a young friend of mine (interesting coincidence, he was a maths student who had dropped out!) Something in my behaviour towards other people had disappointed him. I think I would have had no difficulty in recognising that his disappointment was well-founded, that I had been ungenerous in this instance - but the violence of the reaction literally blew me away. It was like a sudden flare-up of vehement hatred, which died down almost immediately, when it became clear that he hadn't really managed to throw me off balance. (It was close, but I kept that to myself...) I don't know how I got the intuition at the time that he was projecting onto me, duly idealised, unresolved conflicts with his father. This sudden intuition, which was forgotten, didn't stop me from [◊ 69] continuing for years to take on the role of father with the same conviction, without being the least bit suspicious. With, of course, always the same painful astonishment, not believing my eyes or the rest, when later I saw myself confronted with signs of conflict, insidious or violent.

It was only after six or seven months of intense, solitary work on my parents' lives, which made me see them in an unsuspected light, that I realised how illusory is the role of adoptive parent who would replace (for the better, it's understood!) a real parent who does exist, and who would be declared (if only by tacit agreement) to be “failing”. It's helping others to avoid the conflict where it exists, in their relationship with their father, let's say, and projecting it onto a third person (myself in this case) who is completely alien to it. Ever since this meditation, which took place between August 1979 and March 1980, I've been vigilant about myself, so as not to allow myself to indulge in my unfortunate paternal vocation with my eyes closed. This has not prevented the false situation from recurring (as in my relationship with that student with whom I had to stop working) - but now, I believe, without any connivance on my part.

If I put aside the case of the pupil frustrated in these legitimate expectations, there is no doubt in my mind that in all the other cases where I have been confronted with antagonism in a pupil or ex-pupil, it has been the reproduction of the same archetypal conflict with the father: the Father who is both admired and feared, loved and hated - the Man you have to confront, defeat, supplant, perhaps humiliate... but also the One you secretly want to be, stripping Him of a strength to make it your own - another Self, feared, hated and shunned...

​ (30) The enemy father (2)

It wasn't the great turning point of 1970 that created antagonisms between some ex-students and me, against the backdrop of an idyllic and unclouded past. It merely made visible antagonisms that could hardly be expressed within the more conventional framework of a typical boss-student (or expatron-ex-student) relationship. I suspect that such conflicts must not be uncommon in the scientific milieu, but that they are most often expressed in a more roundabout and less recognisable way than in the relationships in [◊ 70] which I have been involved.

Looking back, I don't have the impression, in the end, that in these relationships with my students, I tended so much to enter into a paternal role - indeed, I can't hang on to a single memory that goes in that direction more or less. As far as I'm concerned(Pour ce qui est de ma personne — translator's note), it seems to me that almost all the energy I invested in a relationship with a pupil was the same energy I also invested in mathematics, and in carrying out a vast programme. In the first period, I can think of only one case where there was an interest in me in the person of a student, in the nature of an affinity or sympathy, which had a strength comparable (if not equal) to that of the mathematical interest. But even in that case, I don't have the impression that I took on a paternal role towards him. As for the influence I may have exerted on him or on other students, at one level or another, that's the kind of thing I didn't pay any attention to in my relationship with my students. (Even today, I tend not to pay attention to it, either with the students who have worked with me in recent years, or even with other people. ) Of course, in all these cases, the relationship between the pupil and myself was by no means ‘symmetrical’, in the sense that at least for the duration of the teacher-pupil relationship (and probably even beyond that, more often than not), the importance that a pupil had in my life was not comparable to the importance that I had to assume in his, nor the psychological forces that the relationship brought into play in my person and in his. Except in the five or six cases where these forces manifested themselves in clearly recognised signs of antagonism, I realise that the nature of the relationships with me of my various pupils and then ex-pupils, over more than twenty years of teaching, remains a total mystery to me! In fact, it's not really my job to fathom these mysteries, but rather that of each of them in their own right. But as long as you're interested in your own person, there may be hotter things to look at than the ins and outs of your relationship with your expatron... Be that as it may, even though I showed no inclination towards my pupils to take on a paternal role, it can't have been uncommon for me to have nonetheless more or less acted as an adopted father to them, given my particular psychic ‘profile’ which I mentioned earlier, and also given the dynamics inherent in a situation where I could not fail to act as an elder, to say the least.

[◊ 71] In any case, in several of the cases I have mentioned, this particular colouring of the relationship between a pupil and myself is not in the slightest doubt in my mind. Outside my professional life there have been many more cases where, with or without connivance on my part, I have visibly acted as an adopted father to younger men or women, attracted by my person and linked to me first of all by mutual sympathy, but by no means by ties of kinship. As for my own children, the paternal fibre in me towards them has been strong, and from an early age they have had an important place in my life. By a strange irony, however, none of my five children accepted the fact that I was their father. In the lives of the four of them that I have come to know closely, especially in recent years, this division in their relationship with me reflects a deep division within themselves; a refusal in particular of everything in them that makes them like me, their father... But this is not the place to explore the roots of this division, which go back to a childhood torn apart, to my childhood and that of my parents; as well as to the mother's childhood and that of her parents. Nor is this the place to measure its effects, in their own lives, or in those of their children...

​ (31) The power to discourage

To conclude this summary tour of the relationships I had in the mathematical world between 1948 and 1970, it remains for me to talk about my relationships with younger mathematicians, more or less beginners and therefore without the status of ‘colleague’ strictly speaking, without my playing the role of ‘boss’ vis-à-vis them. These were young researchers whom I met for a year or two in my seminar at the IHES, or on the occasion of a course or seminar at Harvard or elsewhere, or sometimes through correspondence, for example when I had received a piece of work from a young author for which he or she was looking for comments, and certainly also encouragement.

Dealing with beginning researchers is part of a role that is less apparent than that of ‘patron’ of such students, but just as important, as I have since come to realise. At the time, I didn't realise, as I have for the last six or seven years, that this role represents considerable power for a leading mathematician. First of all, it is the power to encourage, to stimulate, which exists just as much in the case of [◊ 72] work that is visibly brilliant (but perhaps served by clumsiness of presentation or a lack of ‘craft’), as in the case of work that is merely solid; it exists even in the case of work that represents only a very modest contribution, if not negligible or even nil according to the criteria of a senior in full possession of powerful means, proven experience of the subject, and extensive information. The power to encourage is present, provided that the work submitted to us has been written seriously - something that is generally discernible from the very first pages.

And the power to discourage exists just as much, and can be exercised at discretion whatever the work. It's the power that Cauchy used against Galois, and Gauss against Jacobi - it's not new that it exists and that eminent and feared men use it! If history has recorded these two cases, it is because the men who had to pay the price had sufficient faith and confidence to continue on their way, despite the unsympathetic authority of those who were then calling the shots in the mathematical world. Jacobi found a journal in which to publish his ideas, and Galois found the pages of his last letter to act as a ‘diary’.

Today, for an unknown or little-known mathematician, it is certainly more difficult than in the last century to make a name for oneself. And the power of the prominent mathematician is not only psychological, but also practical. He has the power to accept or refuse a work, in other words: to give or refuse his support for a publication. Rightly or wrongly, it seems to me that ‘in my day’, in the 1950s and 1960s, rejection was not a foregone conclusion - if the work presented ‘worthy’ results, it had a chance of finding the support of another eminence. Today, this is certainly no longer the case, as it has become difficult to find even one influential mathematician who will agree to review (with whatever willingness he or she may have) a work in his or her field, when the author has not already acquired a reputation, or is not recommended by a well-known colleague.

It has happened to me, during the last few years, to see influential and brilliant mathematicians make use of their power to discourage and refuse, both with regard to such solid work which obviously had to be done, [◊ 73] and with regard to such large-scale works clearly denoting the power and originality of their authors. On several occasions, the person who used his discretionary power in this way happened to be one of my former students. This is without doubt the most bitter experience I have had in my life as a mathematician.

But I am straying from my point, which was to examine the way in which, at the time when I lent myself with conviction to the role of ‘mathematician in the limelight’, I used the power to encourage and discourage that I had at my disposal. I should add that at the more modest level at which my scientific activity continued after 1970, as one of several lecturers at a provincial university, this power did not cease to exist, either vis-à-vis my students or pupils, or (admittedly rarely) vis-à-vis occasional correspondents. But for my present purpose, it is the first period of my life as a mathematician that is important.

As far as my relationship with my pupils is concerned, from the first one I had to the present day, I think I can say without restriction of any kind that I have done everything in my power to encourage them in the work they have chosen (\(23_{IV}\) ). It must be rare, even today, for the relationship between ‘boss’ and pupil to be any different, especially in the case of a boss who has the means to train brilliant pupils and, with their help, clear vast tracts of land ready for ploughing. It's hard to believe, but it's true, that there is even that extreme case of a prestigious boss who takes pleasure in extinguishing in brilliantly gifted students the mathematical passion that had driven him when he was younger.

But I'm digressing again! It is my relationship with the young researchers who were not my students that we now need to examine. In such relationships, the egotistical forces in the person of the man in the spotlight would be less likely to push him in the direction of encouragement, while the successes of the young stranger approaching him would contribute little or nothing to his own glory. On the contrary, I think that the mere play of egotistical forces, in the absence of genuine benevolence, would almost invariably tend to push in the opposite direction, to use the power to discourage, to refuse. This, it seems to me, is no more and no less than that general law which can be [◊ 74] observed in all sectors of society: that the egotistical desire to prove one's own importance, and the secret pleasure which accompanies its gratification, are generally stronger and more appreciated when the power at one's disposal finds occasion to cause the discomfiture of one's neighbour, or even his humiliation, rather than the reverse. This law expresses itself in a particularly brutal way in certain exceptional contexts, such as war, concentration camps, prisons or psychiatric asylums, or even simply in the all-purpose hospitals of a country like ours... But even in the most everyday contexts, each of us has had occasion to be confronted with attitudes and behaviour that attest to this law. The correctives to these attitudes are first of all cultural correctives, stemming from a consensus, in a given environment, on what is considered to be ‘normal’ or ‘acceptable’ behaviour; they are also forces of a non-egotistical nature, such as sympathy towards a particular person, or sometimes, a spontaneous attitude of benevolence independent even of the person to whom it is addressed. Such benevolence is undoubtedly rare in any environment. As for the cultural corrective in mathematics, it seems to me that it has been considerably eroded over the past two decades. This is certainly the case, at least in the circles I have known.

I am determined to stray from my point, which was not a speech about the century, but a meditation on myself and my relationship with the more or less novice researchers who were not my students. I don't think that the ‘law’ I alluded to found expression in these relationships. For reasons that need not be examined here, it would seem that the egotic forces, just as strong in me as in anyone else, have not taken this path in my life to manifest themselves at the expense of others (apart from a few cases dating back to my childhood). I think I can even say, having had the opportunity to examine the matter, that the basic tone of my disposition towards others is one of benevolence, a desire to help when I can help, to relieve when I can relieve, to encourage when I am in a position to encourage. Even in a relationship as deeply divided as that with this ‘tireless friend’ I've been talking about, I've never been so fatuous that I would have thought (even if unconsciously) of harming him. (I would have had the opportunity to do so [◊ 75], and ‘with the best conscience in the world’, of course.) And I think that in most cases these dispositions of general benevolence (even if they were only a little thin-skinned) also marked my relations in the mathematical world, including with beginning mathematicians who, without being among the students, might have needed my support or encouragement.

I believe that this was the case without exception, at least during the 1950s and into the early 1960s. It seems to me that in those days at least, this benevolence was not limited to visibly brilliant young people like Heisuke Hironaka or Mike Artin (even though they were not yet renowned for their abilities). But it is possible that it faded to a greater or lesser extent during the 1960s, under the influence of egotistical forces. I would be particularly grateful for any testimony on this subject.

My memory only gives me a specific case, which I'm going to talk about, and beyond this case, this famous ‘fog’ which doesn't condense into any other case or specific fact, but rather gives me a certain inner attitude. I used to feel a certain irritation when another mathematician ‘stepped on my toes’ without pretending to ask me anything, as if he was at home, the young white boy! These were probably mostly cases of young people, not really up to speed, who thought they were rediscovering things I'd known for years and even longer, sometimes in very special cases. It didn't happen very often, I don't think, but maybe two, three or four times, I'm not sure. As I've just said, I can only remember one case in point, perhaps because the situation was repeated with the same young mathematician on several occasions, in one form or another. I can say that in every respect this young researcher, whose home university was abroad, was perfectly correct in sending me, who was supposed to be the most up to date, the work he had just done. Each time, I reacted very coolly, for the reason I've given. I can't even say for sure if I was telling him straight out that what he was doing had been known to me for ages, and that for that reason it bothered me that he was publishing it without at least giving me a little bow in the introduction. Of course, if he had been my pupil, this authorial fatuity wouldn't have come into play so much, partly because of a sympathetic relationship that had already been established with the pupil, but also because it was [◊ 76] taken for granted anyway that the pupil's work also contained the boss's ideas, unless otherwise stated! I think the situation must have happened twice, maybe even three times, with this same researcher, and each time I had an equally cool, equally discouraging attitude. If I remember correctly, I never agreed to recommend a work by this researcher for publication in a particular journal, nor to sit on a thesis jury (I think I remember the question being asked). It's almost as if I'd decided to make him my target. The best thing is that his work was always perfectly valid - I think it was carefully written, and I have no reason to suppose that he didn't come up with the ideas he was developing himself, which at the time were still not widely known, and were (more or less) only ‘well known’ to a handful of people in the know, like Serre, Cartier, me and one or two others. What's incomprehensible to me is that this young colleague (he ended up, of course, with a thesis and a well-deserved job) never tired of addressing me, who ‘beat him cold’ every time, and that he apparently never held it against me. Still, I remember the surprise he once expressed at my reticence; he obviously didn't understand what was going on. He would have had a hard time if he'd been waiting for me to explain! He had a beautiful head, a bit like a classic Greek, very youthful - rather soft, peaceful features, evoking an inner calm... Now that I'm trying for the first time to put my finger on the impression that his person and physiognomy gave off, I suddenly realise that he really did look a lot like that ‘indefatigable friend’ of whom I've had occasion to speak; they could have been brothers, this friend of my age in a smiling tone, and this researcher, twenty years younger, more in a slightly serious tone, but by no means sad. It's not impossible that this resemblance played a part, that I projected onto one a disdain that had not found an opportunity to express itself with the other, disarmed as he was by the signs of such a faithful friendship! And indeed, I had to have developed a really thick shell not to be disarmed by the obvious good faith and desire to do the right thing in this young man, who was certainly endearing, and who never tired of coming back to me without my deigning to give him even a smile!

​ (32) The mathematician's ethic

The case I reported yesterday, now that I have finally taken the trouble to write it down in black and white, seems to me to be of considerable significance, greater in some respects than the other three cases (doubtless also typical) reported previously, where forces of fatuity deeply disturbed in me a natural attitude of benevolence and respect. This time, using a position of real power (even though I pretended, like everyone else, to be unaware of this power), I used it to discourage a researcher of good will, and to refuse a work that deserved to be published. That's what we call abuse of power. It is no less blatant for not falling under an article of the penal code. It is fortunate that the situation at the time was less difficult than it is today, so that this researcher was able, without too much difficulty I believe, to get his work published with the support of some colleague more benevolent than myself, and that his career as a mathematician was not seriously disrupted, let alone destroyed, by my abusive behaviour. I'm happy about this in retrospect, but I don't want to see it as an ‘extenuating circumstance’. It's possible that in tougher times, I would have been more careful - but that's just a guess, and has little to do here. All the same, I think I can say that there was no secret malice in me, no desire to do harm caused by the irritation I mentioned. I reacted to that irritation ‘viscerally’, without the slightest hint of criticism of myself, or the slightest inclination to look at what was going on inside me, or even the impact my reaction might have had on the other person's life. I didn't appreciate the power I had, and the thought of any responsibility that went with that power (even if it was only the power to encourage or discourage) never occurred to me during that relationship. It was a typical case of irresponsible behaviour, of the kind you find on every street corner, in the scientific world as elsewhere.

It's possible that the only case of his kind that I can remember was an extreme case, one of several similar ones. What triggers an attitude that is devoid of benevolence is the irritation of vanity, impatient to see ‘the first person in’ arrogate to himself the right to walk into guarded hunting grounds and take some small game that belongs only to the masters of those grounds... This irritation has its own rationalisations, which look nobler, as you can imagine. It's not my modest self that's at stake, no, but the love of art and mathematics, this young man who doesn't even have the excuse [◊ 78] of being brilliant, the clumsy type rather, he's going to ruin everything, woe betide us, if only he could do things better than I can, but the beautiful arrangements I had planned, all gone to pot, you've got to be a bit shameless frankly...! There's a constant undercurrent of the meritocratic leitmotiv: only the very best (such as myself) have the right to work for me, or those who put themselves under the protection of one of those! (As for the less common case where it's actually another great chef who walks in my footsteps, that's a different kettle of fish - one day at a time! ) In the case in point, there was (I have little doubt about this) another force moving in the same direction, entirely unconscious, which had already played a major role in my relationship with the indefatigable friend of my early days: an automatic rejection of a certain type of person, not corresponding to the canons of ‘virility’ that I had taken over from my mother. But this circumstance, which has its significance and its interest for an understanding of myself, is relatively irrelevant for my present purpose: that of finding in myself, in attitudes and behaviours that were mine at the time when I was still part of a certain milieu, the typical signs of a profound deterioration that I see there today.

If this case, which I have just examined, seems to me to be of greater significance than the others in which I lacked benevolence and respect, it is because it is the one in which a certain elementary ethic in the profession of mathematician is violated (24). In the milieu where I was welcomed in my early days, the Bourbaki milieu and those close to Bourbaki, this ethic I want to talk about was generally implicit, but it was nevertheless present, alive, the object (it seems to me) of an intangible consensus. The only person who expressed it to me in clear and unambiguous terms, as far as I can remember, was Dieudonné, probably one of the first times I was his guest in Nancy. He may have returned on other occasions. He obviously felt that it was an important thing, and I must have sensed the importance he attached to it then, to have remembered it even today, thirty-five years later. Simply because of the moral authority of the group of my elders, and Dieudonné, who clearly expressed a consensus of the group at the time, I had to tacitly adopt this ethic, without ever having given it a moment's thought, or understood why it was important. To tell the truth, it never even occurred to me that it might be useful for me to give it [◊ 79] some thought, convinced as I had been for a long time that my parents and myself each represented a perfect embodiment (or very close to it) of an ethical attitude, responsible and all-round, and beyond reproach (25).

Dieudonné didn't give me a long speech - it was no more his style than that of any of his friends in Bourbaki. He must have mentioned it in passing, and as something that was taken for granted. He was simply insisting on one of the simplest rules, seemingly innocuous, which is this: anyone who finds a result worthy of interest must have the right and the opportunity to publish it, on the sole condition that this result is not already the subject of a publication. So even if this result was known to one or more people, as long as they did not take the trouble to put it down in black and white and publish it, so as to make it available to (ahem!) the ‘mathematical community’, any other person (by implication: including the famous ‘first comer’! ) who finds the result by their own means (i.e. whatever their means, points of view and insights, and whether or not they seem ‘narrow’ to people supposedly more in the know than they are...) must be given the opportunity to publish it, according to their own means and insights. I think I remember Dieudonné adding that if this rule was not respected, it opened the door to the worst abuses - it's possible that it was on this occasion and through his mouth that I learned of the historic case of Gauss refusing Jacobi's work, on the pretext that Jacobi's ideas had been known to him for a long time.

This simple rule was the essential corrective to the ‘meritocratic’ attitude that existed in Dieudonné (and in other members of Bourbaki) as well as in myself. Respect for this rule was a guarantee of probity. I am happy to be able to say, from everything that has reached me to date, that this essential probity has remained intact in each of the members of the original Bourbaki group (26). I note that this was not the case for other mathematicians who were part of the Bourbaki group or milieu. It has not remained intact in my own person.

The ethic that Dieudonné spoke to me about in down-to-earth terms died as the ethic of a certain milieu. Or rather, that milieu [◊ 80] itself died at the same time as the probity that was its soul. This probity has been preserved in certain isolated persons, and it has reappeared or will reappear in certain others where it had degraded. Its appearance or disappearance in some of us is one of the crucial episodes in the spiritual adventure of each of us. But the scene on which this adventure unfolds is profoundly transformed. A milieu that had welcomed me, that I had made my own, of which I was secretly proud, is no more. What made it worthwhile died within me, or at least was invaded and supplanted by forces of a different nature, long before the tacit ethics that governed it were openly disavowed in customs and professions of faith. If I have been surprised and offended since then, it has been through deliberate ignorance. What came back to me from this milieu that was once mine had a message to bring me about myself, which I have been happy to avoid until now.

VI HARVEST

Contents

  1. The note - or the new ethic
  2. Silt and spring
  3. My passions
  4. Desire and meditation
  5. Wonder
  6. The urge to return and renew
  7. Beautiful by night, beautiful by day - or Augias' stables
  8. Sports mathematics
  9. Off the merry-go-round!

​ (33) The note - or the new ethic

Of course, a rule of professional conduct only takes on its meaning through an inner attitude, which is its soul. It cannot create the attitude of respect and fairness that it seeks to express; at most, it can contribute to the permanence of such an attitude in an environment where the rule enjoys general consensus. In the absence of an inner attitude, even if the rule is professed by the lips, it loses all meaning and value. No amount of exegesis, no matter how scrupulous or meticulous, could change that.

One of my friends and companions of yesteryear kindly explained to me recently that these days, alas, with the inordinate influx of mathematical production, ‘we’ are absolutely obliged, whether we like it or not, to make a strict selection of the papers that are written and submitted for publication, to publish only a small part of them. He said this with an air of sincere regret, as if he himself were a bit of a victim of this inescapable inevitability - a bit like the air he also used to say that he himself was one of the ‘six or seven people in France’ who decide which articles will be published and which will not. Having become less loquacious with age, I confined myself to listening in silence. There was a lot to say on the subject, but I knew it would be wasted effort. A month or two later I learned that this colleague had refused a few years ago to recommend the publication of a certain note to the CRs, [◊ 81] whose author and theme (which I had suggested to him seven or eight years ago) were close to my heart. The author had spent two years of his life developing this theme, which is admittedly not fashionable (although it still seems so topical to me). I think he has done an excellent job (presented as a 3rd cycle thesis). I wasn't the ‘boss’ of this young researcher, who happens to be brilliantly gifted (I don't know whether he'll continue to apply his gifts in mathematics, given the reception...), and he did his work without any contact with me. But it's also true that there could be no doubt as to the origin of the theme developed; he was in a bad way, poor chap, and without suspecting anything, surely! This colleague went out of his way, at least that's what he did, and I wouldn't have expected any less from him: ‘I'm really sorry, but you understand...’. Two years' work by a highly motivated junior researcher for a three-page note to the CR - how much would it have cost in public money? The absurdity of this huge disproportion between the two is obvious. This absurdity surely disappears if we take the trouble to examine the underlying motivations. Only this colleague and former friend is in a position to fathom his own motivations, just as only I am in a position to fathom mine. But without having to go very far, I'm well aware that it's not the excessive influx of mathematical production, you know, nor the public purse (or the patience of an imaginary ‘unknown reader’ of the CRs) that we're trying to spare...

This same draft note to the CR had already had the honour of being submitted to another of the ‘six or seven people in France...’, who sent it back to the author's ‘boss’, because this mathematics ‘didn't amuse him’ (verbatim!). (The boss, disgusted but cautious, himself in a rather precarious position, preferred on both occasions to back down rather than displease...) Having had the opportunity to discuss the matter with this colleague and ex-student, I learned that he had taken the trouble to read the note submitted carefully and to reflect on it (it must have brought back many memories for him...), and that he had found that some of the statements could have been presented in a more user-friendly way. But he didn't deign to waste his precious time submitting his comments to the person concerned: fifteen minutes from the illustrious man, against two years' work from a young unknown researcher! The maths ‘amused’ him enough to seize this opportunity to get back in touch with the situation studied in the note (which could not fail to arouse in him, as in myself, a rich fabric of [◊ 82] various geometrical associations), to assimilate the description given, and then, without difficulty given his background and his means, to detect the clumsinesses or gaps. He did not waste his time: his knowledge of a certain mathematical situation was clarified and enriched, thanks to two years of conscientious work by a researcher taking his first steps; work that the Master would certainly have been able to do (in broad outline and without demonstrations) in a few days. Having achieved this, we remember who we are - the case has been judged, two years of work by Monsieur Personne is fit for the dustbin...

Some people don't feel a thing when that wind blows - but it still takes my breath away today. I'm sure that was one of the desired effects in this case (given the exquisite form of the refusal), but certainly not the only one. In the same interview, this former friend of mine confided in me, with an air of modest pride, that he only agreed to submit a mark to the CRs when ‘the results stated astonished him, or he didn't know how to demonstrate them’ (27). This is undoubtedly one of the reasons why he rarely published. If he applied his own criteria to himself, he wouldn't publish at all. (It's true that in the situation he finds himself in, he doesn't need to.) He knows everything, and it must be as difficult to surprise him as it is to find something demonstrable that he doesn't know how to demonstrate. (This has only happened to me two or three times in the space of twenty years, and not even in the last ten or fifteen!) He is clearly proud of his ‘quality’ criteria, which make him the champion of the highest standards in mathematics. What I saw was an unfailing self-indulgence, and on more than one occasion an unrestrained contempt for others, behind the appearance of smiling, good-natured modesty. I have also seen that he finds great satisfaction in this.

This colleague's case is the most extreme I've come across among the representatives of the ‘new ethics’. It is no less typical. Here again, both in the incident I have reported and in the profession of faith that rationalises it, there is an absurdity that is Ubuesque, in terms of simple common sense - of such enormous dimensions that this former friend with such an exceptional brain, and surely also many of his colleagues with less prestigious status (who will be content not to ask him to submit a note to the CRs) no longer see it. To see it, you have to at least look at it. When you take the trouble to look at the motivations (and first and foremost your own), [◊ 83] then the absurdities appear in full light, and at the same time they cease to be absurd, by revealing their humble and obvious meaning.

If in recent years it has often been so painful for me to be confronted with certain attitudes and above all certain behaviours, it is surely because I have obscurely discerned in them a caricature pushed to the extreme, to the point of grotesqueness or odiousness, of attitudes and behaviours that had once been mine and that were brought back upon me by some of my former students or friends. More than once I've been triggered by the old reflex to denounce, to fight the ‘evil’ that has been clearly pointed out - but if I've given in to it here and there, it's been with divided conviction. Deep down, I know that fighting means continuing to skate on the surface of things, it means evading the issue. My role is not to denounce, or even to ‘improve’ the world in which I find myself, or to ‘improve’ myself. My vocation is to learn, to know this world through myself, and to know myself through this world. If my life can bring any benefit to myself or to others, it will be insofar as I am faithful to this vocation, insofar as I am in tune with myself. It's time to remind myself of this, to cut short those old mechanisms in me, which here would like to push me to plead a cause (of a certain dead ethic, let's say), or to convince (of the supposedly ‘absurd’ nature of some ethic that has replaced it, perhaps), rather than to probe in order to discover and know, or to describe as a means of probing. In writing the preceding two or three pages, with no more precise aim than to say a few words about the current attitudes that have replaced those of yesterday, I felt constantly on my guard towards myself, in the mood of someone who would be prepared at any moment to cross out with a broad stroke everything he has just written and throw it in the bin! But I'm going to keep what I've written, which isn't false but nevertheless creates a false situation, because I'm involving others more than I'm involving myself. Deep down I felt that I wasn't learning anything by writing, and that's probably what created this unease in me. It's definitely time to get back to thinking in a more substantial way, one that instructs me rather than pretending to instruct or convince others (28).

​ (34) Silt and spring

It seems to me that, for the most part, I have come round to what my relations with other mathematicians of all ages and ranks have been, from the time when [◊ 84] I was part of their world, the world of mathematicians; and at the same time, and above all, to the part I have played, through my own attitudes and behaviour, in a certain spirit that I see there today, and which is surely not of yesterday. In the course of this reflection, or journey to put it more accurately, I came across four situations which struck me as typical of certain attitudes and ambiguities in myself, where spontaneous dispositions of benevolence and respect towards others were disrupted, if not totally swept away, by egotistical forces, and above all (in three of these cases at least) by fatuity. This fatuity was based above all on the supposed superiority conferred on me by a certain cerebral power and the inordinate investment I made in my mathematical activity. It found confirmation and support in a general consensus that valued, almost without reservation, this cerebral power and this inordinate investment.

It is the last of the situations examined, that of the ‘unlearned youngster who stepped on my toes’, which seems to me to be the most important of the four for my present purpose. The first three are typical of me, or of certain aspects of me, at a certain time (in a certain context too, it's true) - but, as I've had occasion to say over and over again, I don't consider them in any way typical of the milieu to which I belonged. Nor do I think they are typical of the current mathematical milieu in France, let's say - it's probable that the kind of chronic bewilderment that characterised my relationship with ‘l'ami infatigable’, for example, is as uncommon nowadays as it must have been then. My attitude and behaviour in the case of the ‘young misfit’, on the other hand, is typical of what happens every day in the mathematical world, wherever you look. It is the attitude of kindness and respect shown by the influential mathematician towards the young stranger that becomes the rare exception, when the said stranger does not have the good fortune to be his pupil (and yet...), or the pupil of a colleague of comparable status and recommended by him. This is no doubt what I was already aware of in the aftermath of my ‘awakening’ in 1970, which had loosened silent tongues - but the first-hand accounts I heard at the time remained remote for me, because they didn't directly concern me or my dearest friends in my milieu. I was affected more than superficially from the moment (around 1976) when the echoes [◊ 85] that came back to me, or the events I witnessed, had as their protagonists some of these friends, or even ex-students who had become important, and even more so when those who were the target of malice were people I knew well, pupils on more than one occasion (pupils ‘after 1970’, it goes without saying!), whose fate therefore affected me. In some cases, there was no longer any doubt that the lack of benevolence, or even an attitude of ostentatious contempt, was reinforced, at the very least, if not provoked, by the mere fact that a young researcher was my pupil, or that he was taking the risk (without necessarily being my pupil) of doing what my friends of yesteryear and other colleagues also like to call ‘grothendieckeries’...

The ‘young misfit’ wrote to me again in the early 1970s, very courteously asking me (although he was under no obligation to ask me anything at all!) if I had no problem with him publishing a proof he had found for a theorem he had been told I had written, but which had never been published. I remember that I replied to him in the same bad-tempered way as before, without saying yes or no I think, and implying, without knowing his demonstration (which he was of course prepared to send me but which I didn't care about, busy as I was with my militant duties!), that it would certainly not add anything to mine (although it would at the very least have benefited from being written down in black and white and available to the mathematical public, as well as the statement itself!) This just goes to show the extent to which this famous ‘awakening’ was still superficial, without any impact on certain behaviours rooted in a fatuity and in ‘meritocratic’ attitudes, which I was no doubt denouncing at the same time in well-intentioned articles in Survivre et vivre, in speeches in public debates and so on.

This is a very concrete answer to a question I left open earlier. I might as well admit this humble truth, that such attitudes of fatuity are by no means overcome ‘once and for all’ in my person, and I doubt that they ever will be, except at my death. If there has been a transformation, it is not through the disappearance of vanity, but through the appearance (or reappearance) of curiosity about myself and the true nature of certain attitudes, behaviours, etc., in myself. It is through this curiosity that I have become somewhat sensitive to the manifestations of vanity in myself. This profoundly modifies a certain [◊ 86] inner dynamic, and thereby modifies the effects of ‘vanity’; that is to say, of that force which often pushes me to conceal or counterfeit the healthy and fine perception I have of reality, for the purpose of enlarging my person and putting myself above others while pretending the opposite.

Perhaps a reader will feel baffled, as I once did, by the apparent contradiction between the insidious and tenacious presence of vanity in my life as a mathematician (which he may also have glimpsed at times in his own), and what I call my love, or passion, for mathematics (which perhaps also echoes in his own experience of mathematics, or of some other person or thing). If he is baffled indeed, he has within him everything he needs to get back in touch (as I once did) with the reality of things themselves, which he can know at first hand, rather than spinning around like a squirrel trapped in an endless cage of words and concepts.
Will someone who sees muddy water say that water and mud are one and the same thing? To know the water that is not mud, all you have to do is go up to the source and look and drink. To know the mud that is not water, all you have to do is go up to the bank, dried by the sun and the wind, and detach a ball of grainy clay in your hand. Ambition and vanity can more or less regulate the proportion of one's life devoted to a particular passion, such as mathematical passion, and can make it all-consuming, if the returns satisfy them. But the most devouring ambition is powerless in itself to discover or to know the least of things - quite the contrary! In the moment of work, when little by little an understanding begins, takes shape, deepens; when in a confusion little by little an order appears, or when what seemed familiar suddenly takes on unusual, then disturbing aspects, until finally a contradiction bursts out and overturns a vision of things that seemed unchanging - in such work, there is no trace of ambition, or vanity. What then leads the dance is something that comes from much further away than the ‘I’ and its hunger to constantly expand (even if it is with ‘knowledge’) - from much further away than our person or even our species.

This is the source, which is within each one of us.

​ (35) My passions

[87] Three great passions have dominated my adult life, alongside other forces of a different nature. I have come to recognise in these passions three expressions of the same deep-seated drive; three paths that the drive for knowledge in me has taken, among an infinite number of paths open to it in our infinite world.

The first to manifest itself in my life was my passion for mathematics. At the age of seventeen, when I left secondary school, I let go of a simple inclination and turned it into a passion, which directed the course of my life for the next twenty-five years. I ‘knew’ mathematics long before I knew the first woman (apart from the one I knew from birth), and today in my middle age I can see that it still hasn't burnt out. She no longer runs my life, any more than I claim to run it. Sometimes it slumbers, sometimes to the point where I think it's extinguished, only to reappear unannounced, as fiery as ever. It no longer devours my life as it once did, when I gave it my life to devour. It continues to leave a deep imprint on my life, like the imprint in a lover of the woman he loves.
The second passion in my life was the quest for a woman. This passion often presented itself to me as the quest for a companion. I wasn't able to distinguish one from the other until the latter came to an end, when I realised that what I was pursuing was nowhere to be found, or else: that I was carrying it within myself. My passion for women only really developed after my mother's death (five years after my first love affair, from which a son was born). It was then, at the age of twenty-nine, that I started a family, from which three other children were born. My attachment to my children was originally an indissoluble part of my attachment to my mother, a part of the power emanating from the woman who attracted me to her. It's one of the fruits of this passion for love.

I didn't experience the presence of these two passions in me as a conflict, either at the beginning or later. I must have had an obscure sense of the profound identity of the two, which became clear to me much later, after the third appeared in my life. Yet the effects of the two passions on my life were bound to be very different. The love of mathematics drew me into a certain world, that of mathematical objects, which surely has [◊ 88] its own ‘reality’, but which is not the world in which human life takes place. The intimate knowledge of mathematical things taught me nothing about myself, as much as anything, and even less about others - the impulse of discovery towards mathematics could only distance me from myself and from others. Two or more people can sometimes come together in the same impulse, but this is communion on a superficial level, which in fact distances each of them from themselves and from others. This is why my passion for mathematics has not been a force for maturation in my life, and I doubt that such a passion can encourage maturation in anyone (29). If I gave this passion such a disproportionate place in my life for a long time, it was surely also precisely because it allowed me to escape the knowledge of conflict and the knowledge of myself.

The sex drive, on the other hand, whether we like it or not, launches us straight into the encounter with others, and straight into the crux of the conflict in ourselves and in others! The quest for ‘companionship’ in my life was the quest for bliss without conflict - it wasn't the drive for knowledge, the drive for sex, as I liked to believe, but an endless flight from the knowledge of the conflict in the other and in myself. (This was one of the two things I had to learn, so that this illusory quest would come to an end, and the anxiety that accompanies it like its inseparable shadow...) Fortunately, no matter how much we run away from conflict, sex quickly brings us back to it!

One day I gave up trying to deny the teaching that the conflict stubbornly brought me, through the women I loved or had loved, and through the children born of those loves. When I finally began to listen and learn, and for years to come, it turned out that everything I learned was from the women I had loved or loved (30). Until 1976, at the age of forty-eight, it was the quest for women that was the only great maturing force in my life. If this maturation only took place in the years that followed, in other words over the last seven years, it's because I protected myself (as I had learned to do from my parents and the people around me) by all the means at my disposal. The most effective of these means was my investment in my passion for mathematics.
The day the third great passion appeared in my life - a [◊ 89] certain night in October 1976 - the great fear of learning vanished. It was also the fear of simple reality, of humble truths about myself first and foremost, or about people who were dear to me. Strangely enough, I had never perceived this fear in myself before that night, at the age of forty-eight. I discovered it on the very night that this new passion, this new manifestation of the passion to know, appeared. It took the place, so to speak, of the fear that had finally been recognised. For years I had seen this fear clearly in others, but by some strange blindness I had not seen it in myself. The fear of seeing prevented me from seeing this very fear of seeing! Like everyone else, I was strongly attached to a certain image of myself, which for the most part had not changed since I was a child. The night I'm talking about is also the night when, for the first time, that old image collapsed. Other images like it followed, holding on for a few days or months, or even a year or two, thanks to stubborn forces of inertia, only to collapse in their turn under scrutiny. The laziness of looking often delayed such a new awakening - but the fear of looking never reappeared. Where there is curiosity, there is no room for fear. When I am curious about myself, there is no more fear of what I am going to find than when I want to know the final word of a mathematical situation: there is then a joyful expectation, impatient at times and yet obstinate, ready to welcome whatever comes its way, foreseen or unforeseen - a passionate attention on the lookout for unequivocal signs that make it possible to recognise the true in the initial confusion of the false, the half-true and the perhaps.

In curiosity about ourselves, there is love, untroubled by any fear that what we are looking at might not be what we would like to see. And to tell the truth, love for myself had silently blossomed in the months leading up to this night, which is also the night when this love took on an active, enterprising form, if you like, ruthlessly shaking up costumes and sets! As I said, other costumes and sets soon reappeared as if by magic, to be shaken up in their turn, without invective or gnashing of teeth...

The manifestations of this new passion in my life over these last seven years have come to seem to me like the moving up-and-down of waves following one another, like the breaths of a vast and peaceful respiration [◊ 90]. This is not the place to try to trace its sinuous and changing line, or that, in counterpoint, of the manifestations of mathematical passion. I have given up trying to regulate the course of one or the other - it is rather this double movement of one and the other which today regulates the course of my life - or to put it better, which is its course.

In the months that preceded the appearance of the new passion - months of gestation and fulfilment - the woman's quest began to change its face. It began to separate itself from the anxiety with which it had been imbued, like a ‘breath’ that had freed itself from the oppression that had weighed it down, and regained its amplitude and rhythm. Or like a fire that has been smouldering, half-stifled for lack of an escape route, and which, under a breath of fresh air, suddenly bursts into crackling, agile, lively flames!

The fire burned to satiation. A hunger that seemed unquenchable was satisfied. For the last two or three years, it seems that this quest has been consumed without a trace of ashes, leaving the field open to the song and counter-song of two passions. One, the passion of my youth, served for thirty years to separate me from a disowned childhood. The other is the passion of my middle age, which led me to rediscover both the child and my childhood.

​ (36) Desire and meditation

The night I mentioned, when a new passion took the place of an old fear that had vanished forever, was also the night I discovered meditation. It was the night of my first 'meditation', which came under the pressure of an urgent need, after I had been overwhelmed by waves of anxiety in the days before. Like all anxiety, perhaps, it was a 'take-off anxiety', insisting that I take off from a humble and obvious reality about myself, from an image of myself that was forty years old and never questioned by me. Surely there must have been a great thirst for knowledge, alongside considerable forces of flight, and a desire to escape anguish, to be at peace as before. So there was intense work, which continued for several hours until it came to an end, without me yet knowing the meaning of what was happening, let alone where I was going. In the course of this work, the red herrings were recognised one after the other; or to put it better, it was this work that made [◊ 91] these red herrings appear one by one, each in the guise of an intimate conviction that I finally took the trouble to write down in black and white as if to get a better grasp of it, whereas until then it had remained in a propitious blur. I wrote it down quite happily, without mistrusting it in the slightest; it must surely have had something to seduce me - in the mood then of someone who doubts nothing, and for whom the mere fact of having written down an informal conviction in black and white was the irrefutable sign of its authenticity, the proof that it was well-founded. If I hadn't had this indiscreet, not to say indecent, desire to know, I mean, I would have stopped every time at this happy ending, and it was with this happy ending that the stage ended. And then, woe is me! I had a whim, God knows how and why, to take a closer look at what I had just written to my complete satisfaction: it was there in black and white, all I had to do was reread it! And as I reread it carefully, naively, I sensed that there was something a little bit wrong, that it wasn't quite so clear! Then, taking the trouble to look a little more closely, it became clear that it wasn't that at all, that it was all bogus, in other words, that I'd just been led astray! Each time, this partial discovery came as a famous surprise: "Wow, that's really something", a joyful surprise that rekindled my thinking with a surge of new energy. We're going to get to the bottom of this, and I'm sure it's coming no later than now, so all we have to do is keep up the momentum! All we want to do is believe, that's what it must be this time. We're going to write it down anyway, as a matter of conscience, and it's even a pleasure to write down such judicious and well-intentioned things, you'd really have to be wrong-headed not to agree, such obvious good faith, you can't do better than that, it's perfect as it is!

That was the new end of the stage, the new happy ending, on which I would have paused contentedly, if it hadn't been for the naughty boy who was as polite as they come, and who once again got into mischief, daring, incorrigible as ever, to stick his nose into this last happy ending. There was no stopping him, he was off again for another stage!

And so, for four hours, one stage followed another [◊ 92], like an onion whose layers I had peeled off one after the other (that's the image that came to me at the end of that night), to arrive at the end of the ends at the heart - at the simple and obvious truth, a truth that was staring me in the face, and yet which for days and weeks (and my whole life, in fact) I had managed to hide under this accumulation of ‘layers of onion’ hiding one behind the other.

The appearance at last of the humble truth was an immense relief, an unexpected and complete deliverance. I knew at that moment that I had hit the nerve of anguish. The anguish of the last five days was well and truly resolved, dissolved, transformed into the knowledge that had just formed within me. The anguish had not only disappeared from my sight, as it had throughout the meditation, and several times during the previous five days; and the knowledge into which it had been transformed was in no way in the nature of an idea, of a concession that I would have made, let's say, in order to be left alone and at peace (as had happened to me here and there during the same night); it was not an external thing that I would then have adopted or acquired in order to add it to my person. It was knowledge in the full sense of the word, first-hand, humble and obvious, which was now part of me, just as my flesh and blood are part of me. What's more, it was formulated in clear and unequivocal terms - not in a long speech, but in a silly little sentence of three or four words. This formulation had been the final stage of the work that had just been carried out, which remained ephemeral, reversible as long as this final step had not been taken. Throughout this work, the careful, even meticulous formulation of the thoughts that were being formed, the ideas that were being presented, had been an essential part of this work, in which each new departure was a reflection on the stage I had just been through, which was known to me through the written testimony I had just given (with no possibility of concealing it in the mists of a failing memory!).

In the minutes that followed the moment of discovery and deliverance, I also knew the full significance of what had just happened. I had just discovered something even more precious than the humble truth of the last few days. That something was the power within me, provided I [◊ 93] was interested, to get to the bottom of what was going on within me, of any situation of division, of conflict - and by the same token the ability to resolve entirely, by my own means, any conflict within me of which I had become aware. The resolution does not come about through the effect of some grace, as I had tended to believe in previous years, but through intense, obstinate and meticulous work, making use of my ordinary faculties. If there is such a thing as ‘grace’, it is not in the sudden and definitive disappearance of a conflict within us, or in the appearance of an understanding of the conflict that comes to us ready-made (like the chickens in the land of plenty!) - but it is in the presence or appearance of this desire to know (31). It was this desire that had guided me to the heart of the conflict in the space of a few hours - just as the desire for love leads us unfailingly to the path that leads to the innermost depths of the woman we love.

Whether we're talking about self-discovery or mathematics, in the absence of desire, all so-called ‘work’ is just a sham that leads nowhere. At its best, it keeps the person who indulges in it ‘beating around the bush’ endlessly - the contents of the bush are reserved for those who are hungry enough to eat! Like everyone else, I sometimes find that desire and hunger are absent. When it comes to the desire for self-knowledge, then my knowledge of myself and the situations I'm involved in remains inert, and I act not with full knowledge of the facts, but at the whim of simple inveterate mechanisms, with all the consequences that implies - a bit like a car driven by a computer, not by a person. But whether it's meditation or mathematics, I wouldn't dream of pretending to ‘work’ when there's no desire, when there's no hunger. That's why I've never meditated for even a few hours, or done maths for even a few hours (32), without learning something; and most often (if not always) something unforeseen and unpredictable. This has nothing to do with any faculties that I might have that others don't, but simply stems from the fact that I don't pretend to work without really wanting to. (It's the strength of this ‘desire’ that in itself also creates the requirement I've talked about elsewhere, which means that when it comes to work you're not satisfied with just a little, but are only satisfied once you've reached the end of your understanding, however humble it may be). Where it is a question of discovery, work without desire is nonsense and [◊ 94] sham, just as much as making love without desire. To tell the truth, I haven't known the temptation to waste my energy pretending to do something I have no desire to do, when there are so many exciting things to do, if only to sleep (and dream...) when it's time to sleep.

It was on that same night, I think, that I realised that the desire to know and the power to know and discover are one and the same thing. As long as we trust it and follow it, it is desire that leads us to the heart of the things we want to know. And it is desire that makes us find, without even having to look for it, the most effective method for knowing these things, and the one that best suits us as individuals. As far as mathematics is concerned, it would seem that writing has always been an indispensable means, regardless of who is ‘doing maths’: doing maths is first and foremost writing (33). The same is undoubtedly true of any work of discovery in which the intellect plays a major role. But this is certainly not necessarily the case with ‘meditation’, by which I mean the work of self-discovery. In my case, however, and up to now, writing has been an effective and indispensable means of meditation. As in mathematical work, it is the material support that sets the pace of reflection, and serves as a reference point and rallying point for an attention that otherwise tends to scatter to the four winds in my case. Writing also gives us a tangible trace of the work that has just been done, to which we can refer at any time. In long-term meditation, it's often useful to be able to refer back to written records of a particular moment in meditation in the days before, or even years before.

Thought, and its meticulous formulation, therefore play an important role in meditation as I have practised it up to now. However, it is not limited to the work of thought alone. Thought alone is powerless to grasp life. It is effective above all in detecting contradictions, often enormous to the point of grotesqueness, in our vision of ourselves and our relationships with others; but often it is not enough to grasp the meaning of these contradictions. For those driven by the desire to know, thought is often a useful and effective instrument, even indispensable, as long as we remain aware of its limits, which are quite obvious in [◊ 95] meditation (and more hidden in mathematical work). It is important that thought should know how to fade away and disappear on tiptoe at sensitive moments when something else appears - in the form perhaps of a sudden and profound emotion, while the hand perhaps continues to run over the paper to give it at the same moment a clumsy and stammering expression....

​ (37) Wonder

This retrospective on the discovery of meditation came about entirely unexpectedly, almost against my better judgement - it wasn't at all what I set out to examine when I began. I wanted to talk about wonder. This night, so rich in so many things, was also rich in wonder at these things. Already in the course of the work, there was a kind of incredulous wonder at every new red herring that came to light, like a crude costume sewn with thick white thread that I had been willing - it was scarcely believable - to take for real as seriously as possible! Many times since then, in the years that followed, I've found myself marvelling as I did on that first night of meditation, at the enormity of the facts I was discovering and the crudeness of the subterfuges that had made me ignore them until then. I began to discover the unsuspected world I carry within me, a world that over the days, months and years has revealed itself to be prodigiously rich. On that first night, however, I had more to marvel at than vaudeville episodes. It was the night when for the first time I got back in touch with a forgotten power that had been sleeping inside me, the nature of which still escaped me, except that it is a power, and which is at my disposal at any time.

And the previous months had already been filled with a silent wonder at something I'd carried within me, probably for as long as I could remember, with which I'd only just made contact again. I felt this thing not as a power, but rather as a secret sweetness, as a beauty that was both very peaceful and troubling. Later, in the exultation of discovering my long-ignored power, I forgot those months of silent gestation, to which only a few scattered poems bore witness - love poems, which perhaps would have stood out more often than not in the midst of my meditation notes...

It was only years later that I remembered those times [◊ 96] of wonder at the beauty of the world and the beauty I felt resting within me. I knew then that this gentleness and beauty I had felt within me, and this power I discovered shortly afterwards that profoundly changed my life, were two inseparable aspects of one and the same thing.

And I also see now that the gentle, collected, silent aspect of this multiple thing that is creativity in us, is expressed spontaneously through wonder. And it is also in the wonder of an indescribable beauty revealed by the beloved that the man knows the woman he loves and that she knows him. When wonder in the thing explored or in the being loved is absent, our embrace with the world is mutilated of the best that is in it - it is mutilated of what makes it a blessing for ourselves and for the world. An embrace that is not one of wonder is an embrace without strength, a mere reproduction of a gesture of possession. It is powerless to engender anything other than yet more reproductions, bigger or fatter or thicker perhaps, who cares, never a renewal (34). It is when we are children and ready to marvel at the beauty of things in the world and in ourselves that we are also ready to renew ourselves, and ready as supple and docile instruments in the hands of the Worker, so that through His hands and through us beings and things may perhaps be renewed.

I well remember that in this group of unpretentious friends who for me represented the mathematical milieu at the end of the 1940s and in the years that followed - a milieu that was sometimes noisy and self-assured, where a somewhat peremptory tone was not so uncommon (but without any hint of smugness) - in this milieu there was always room for wonder. The one in whom wonder was most visible was Dieudonné. Whether he was giving a talk, or just listening, when the crucial moment came and a sudden breakthrough opened up, Dieudonné would be seen beaming and ecstatic. It was pure, infectious, irresistible wonder - where every trace of ‘me’ had disappeared. As I recall him now, I realise that this wonder itself was a power, that it exerted an immediate action all around him, like a radiance from which he was the source. If ever I saw a mathematician use a powerful and elementary ‘power of encouragement’, it was him! I never thought about it before this moment, but I remember now that it was in this way that he had already welcomed my very first results in Nancy, [◊ 97] solving questions that he had posed with Schwartz (on the spaces (F) and (LF)). They were very modest results, nothing brilliant or extraordinary certainly, one could say that there was nothing to marvel at. Since then I've seen things of a completely different scale rejected by the unanswerable disdain of colleagues who think they're great mathematicians. Dieudonné was in no way encumbered by such pretensions, justified or not. There was nothing of the sort that prevented him from being delighted by even the smallest things.

There is a generosity in this capacity for delight that is a blessing for those who are willing to let it blossom in themselves and for those around them. This benefit is exercised without the intention of pleasing anyone. It is as simple as the fragrance of a flower, or the warmth of the sun.

Of all the mathematicians I have known, it is in Dieudonné that this ‘gift’ appeared to me in the most dazzling way, the most communicative, the most active too perhaps, I cannot say (35). But this gift was not absent in any of the mathematician friends I liked to meet. It found the opportunity to manifest itself, perhaps in a more restrained way, at any time. It manifested itself every time I came up to one of them to share something I had just found and which had delighted me.

If I have experienced frustration and sorrow in my life as a mathematician, it is above all in not finding, in some of those I have loved, that generosity I had known in them, that sensitivity to the beauty of things, ‘small’ or ‘great’; as if what had made up the quivering life of their being had died out without a trace, smothered by the complacency of someone for whom the world is no longer beautiful enough for him to deign to rejoice in it.

There was also, of course, the other pain of seeing one of my friends of yesteryear treat another of my friends of today with condescension or contempt. But this pain is inflicted by the same closure, basically. Anyone who is open to the beauty of a thing, however humble, when he has felt that beauty, cannot help but also feel respect for the person who conceived or made it. In the beauty of something made by human hands, we feel the reflection of a beauty in the person who made it, of the love he put into making it. When we feel that beauty, that love, there can be no condescension or disdain in us, any more than there can be condescension or disdain [◊ 98] for a woman, in a moment when we feel her beauty, and the power in her of which that beauty is the sign.

​ (38) The urge to return and renew

The rapture that radiated from time to time in Dieudonné's person must have touched something deep and strong in me, so that the memory comes back to me now with such intensity, such freshness, as if I'd just witnessed it again just now (although it's been nearly fifteen years since I've had much of an opportunity to meet Dieudonné, apart from once or twice in a flash). Of course, I wasn't paying any particular attention to it on a conscious level - it was just a slightly touching, at times almost comical, feature of the expansive personality of my colleague, elder and friend. What was important to me, however, was to have found in him the perfect collaborator, a dream collaborator I might say, to lay down in black and white with meticulous care, loving care, what was to serve as the foundation for the vast prospects I saw opening up before me. It is only at this moment when I mention both that the link suddenly appears to me: what made Dieudonné the ideal servant for a great task, whether within Bourbaki or in our collaboration on another great foundation project, was his generosity, the absence of any trace of vanity, in his work and in the choice of his major investments. I have constantly seen him take a back seat to the tasks he has made himself the servant of, lavishing inexhaustible energy on them without seeking any return. There is no doubt that, without seeking anything in return, he found in his work and in the very generosity he put into it a fulfilment and fulfilment that all those who knew him must have felt.

The rapture of discovery that I have so often felt radiating from his person, is immediately associated in me with a similar rapture, which I happened to witness in a very young child. There are two memories that come flooding back to me - both of which take me back to my very young daughter. In the first image, she must be a few months old, she must have just started crawling. She must have dragged herself from the piece of grass where she was sitting to a gravel path. She was discovering the little gravels in silent ecstasy - and in action, grabbing them with her hands and putting them in her mouth! In the other picture she must have been a year or two old, someone had just thrown pellets into a goldfish bowl. [99] The fish did their best to swim towards them, mouths wide open, to swallow the tiny yellow crumbs suspended in the air as they slowly sank into the water in the bowl. The little girl had never realised before that fish eat the way we do. It was like a sudden dazzle for her, expressed in a cry of pure delight: ‘Look, Mummy, they're eating! There was indeed much to marvel at - she had just discovered in a sudden flash a great mystery: that of our kinship with all other living beings...

In the rapture of a little child, there is a communicative force that escapes words, a force that radiates from him and acts on us, even though we do our best, more often than not, to hide from it. In moments of inner silence, we feel this force present in the child at all times. Only at certain moments is its action stronger than at others. It is in newborn babies, in the first days and months of life, that this sort of ‘force field’ around the child is most powerful. More often than not, it remains sensitive throughout childhood, unravelling over the years until adolescence, when there often seems to be no trace of it left. Yet it can be found radiating around people of all ages, at special moments for some, or for others as a kind of breath or halo that surrounds them at all times. I was very lucky to know such a person when I was a child, a man who has now passed away.

I'm also thinking of that other strength, or power, that you can sometimes feel radiating from a woman, especially at times when she is fulfilled in her body, in communion with it. The word that often comes to mind is ‘beauty’, which evokes one aspect of it. It's a beauty that has nothing to do with canons of beauty or so-called ‘perfection’, it's not the privilege of youth or maturity. Rather, it is the sign of a deep harmony within the person. This agreement often remains fragmentary, and yet it manifests itself in this radiance, a sign of power. It's a force that draws us towards the centre from which it emanates - or rather, it calls within us a profound impulse to return to the body of the Mother-Woman from which we emerged at the dawn of our lives. Its action is sometimes irresistibly powerful, overwhelming when it emanates from the woman we love. But for those who do not deliberately close themselves off to it, [◊ 100] it is perceptible in any woman who allows this beauty, this profound harmony, to blossom within her.

The strength that radiates from the child is closely related to this strength that emanates from the woman who loves herself in her body. The one is constantly born of the other, just as the child is constantly born of the Mother. But the nature of the force of childhood is neither attraction nor repulsion. The humble and discreet action that this force exerts on those who do not shrink from it is one of renewal.

​ (39) Beautiful by night, beautiful by day - or Augias' stables

The memory of wonder in one of my children is from the very late 1950s and early 1960s. If I have no similar memory for the other children who were born later, it may be that my own capacity for wonder had dulled, that I had become too distant to share in the delight of one of my children, or to witness it at all.

I have never yet thought of tracking the vicissitudes of this capacity in my life, from my childhood to the present day. Surely there would be a common thread there, a ‘detector’ of great sensitivity. If I've never thought of following this thread, it's surely because this ability is of such a humble nature, almost insignificant in appearance, that the idea would hardly have occurred to me to pay any particular attention to it, absorbed as I was in discovering and probing what I used to call ‘the great forces’ in my life (which continue to manifest themselves even today). And yet, this capacity to appear so humble provides a sign, of all signs, of the presence or absence of the rarest and most precious ‘force’ within us...

Throughout my adult life, I have never been entirely cut off from this strength. However arid my life may have been, in love I rediscovered the wonder of the child, the delight of discovery. Through many deserts, the passion of love remained the living, vigorous link with something I had left behind, an umbilical cord that silently continued to nourish me with warm, generous blood. And for a long time, too, wonder in the woman I loved was inseparable from wonder in the new beings she gave birth to - these brand-new, infinitely delicate, intensely alive beings who attested to and inherited her power.

[◊ 101] But my main purpose here is to follow the vicissitudes of this ‘force of innocence’ through my life as a mathematician, during the period when I was part of the ‘world of mathematicians’, from 1948 to 1970. Surely, wonder has never permeated my mathematical passion to the extent that it does the passion of love. Strangely enough, if I try to remember a particular moment of delight or wonder in my mathematical work, I can't find any! My approach to mathematics, from the age of seventeen when I began to invest myself fully in it, has been to set myself big tasks. Right from the start, they were always tasks of ‘tidying up’, of cleaning up. I saw an apparent chaos, a confusion of heterogeneous things or sometimes imponderable mists, which obviously had to have a common essence and conceal an order, a hidden harmony that had to be brought out through patient, meticulous and often lengthy work. It was often a job of mopping and sweeping, for the big jobs that already absorbed considerable energy, before coming to the finishing touches with a feather duster, which I was less passionate about but which also had their charm and, in any case, an obvious usefulness. There was an intense satisfaction in the day-to-day work, seeing little by little the order that had been guessed at emerge, which always turned out to be more delicate, with a richer texture than what had been glimpsed and guessed at. The work was constantly full of unforeseen episodes, most often arising from the examination of what might have seemed an infinitesimal detail that had been neglected until then. Often the fine-tuning of a particular ‘detail’ threw unexpected light on work done years before. Sometimes, too, it led to new insights, which became the subject of another ‘major task’.

So, in my mathematical work (apart from the ‘difficult year’ around 1954 that I've already mentioned), there was constant suspense; my attention was constantly kept on the edge of my seat. Fidelity to my ‘tasks’ prevented me from escaping too far, and I gnawed my teeth in impatience to have reached the end of them all and to be embarking at last on the unknown, the real unknown - even though the scale of these tasks had already become such that to bring them to a successful conclusion, even with the help of the good people who eventually came to the rescue, the rest of my days would not have been enough!

My main guide in my work was the constant search for a perfect coherence, a complete harmony that I divined behind the [◊ 102] turbulent surface of things, and which I strove patiently to draw out, never tiring of it. It was a keen sense of 'beauty', surely, that was my flair and my only compass. My greatest joy was not so much to contemplate it when it appeared in the full light of day, as to see it gradually emerge from the cloak of shadow and mist in which it was constantly hiding. Of course, I didn't stop until I had succeeded in bringing it into the clearest light of day. Sometimes I experienced the fullness of contemplation, when all the audible sounds contribute to the same vast harmony. But even more often, what was brought out into the open immediately became the motivation and the means for a new plunge into the mists, in pursuit of a new incarnation of the One who remained forever mysterious, unknown - constantly calling out to me, to know Her again...

Dieudonné's pleasure and delight was above all, it seems to me, in seeing the beauty of things manifest themselves in full light, and my joy was above all in pursuing it in the dark recesses of the mists and the night. This is perhaps the profound difference between Dieudonné's and my approach to mathematics. The sense of the beauty of things, for a long time at least, must have been no less strong in me than in Dieudonné, although it may have dulled during the 1960s, under the influence of fatuity. But it would seem that my perception of beauty, which manifested itself in Dieudonné as wonder, took on different forms in me: less contemplative, more enterprising, and also less manifest in terms of the emotion felt and expressed. If this is the case, then my aim would be to follow the vicissitudes of my openness to the beauty of mathematical things, rather than the mysterious 'gift of wonder'.

​ (40) Sports mathematics

It is quite clear that openness to the beauty of mathematical things never entirely disappeared in me, even in the 1960s until 1970, when fatuity gradually took an increasing place in my relationship to mathematics and to other mathematicians. Without a minimum of openness to the beauty of things, I would have been incapable of ‘functioning’ as a mathematician, even on the most modest of terms - and I doubt that anyone could do any useful work in mathematics if this sense of beauty did not remain alive in them to some extent. It is not so much, it seems to me, a [◊ 103] supposed ‘brain power’ that makes the difference between one mathematician and another, or between one work and another by the same mathematician; but rather the quality of finesse, of greater or lesser delicacy of this openness or sensitivity, from one researcher to another or from one moment to another in the same researcher. The most profound, the most fruitful work is also that which demonstrates the most delicate sensitivity for apprehending the hidden beauty of things (36).

If this is the case, then this sensitivity must have remained alive in me right up to the end, at least at times, since it was at the end of the 1960s 1 that I began to catch a glimpse of the most hidden and mysterious mathematical thing I had ever discovered - the thing I called ‘pattern’. It is also the thing that has held the greatest fascination for me in my life as a mathematician (apart from certain reflections in recent years, which are intimately linked to the reality of patterns). There is no doubt that if my life had not suddenly taken an entirely unexpected course, taking me far away from the serene world of mathematical things, I would have ended up following the call of this powerful fascination, leaving behind the ‘tasks’ that had until then kept me prisoner!
Perhaps I can say that, in the solitude of my work room, my sense of beauty remained unchanged until my first ‘awakening’ in 1970, unaffected by the fatuity that so often marked my relationships with my peers? A certain ‘flair’ must even have been refined over the years, through daily and intimate contact with mathematical things. The intimate knowledge that we can have of things, which sometimes allows us to go beyond what we know at the moment and penetrate further into knowledge - this knowledge or this maturity, and this ‘flair’ which is its most visible sign, is closely related to openness to the beauty and truth of things. It fosters and stimulates such openness, and it is the sum and fruit of all the moments of openness, of all the ‘moments of truth’ that have gone before.

What remains for me to examine, then, is the extent to which a spontaneous sensitivity to beauty was disturbed to a greater or lesser extent at the moments when it had the opportunity to manifest itself in my relationship with this or that colleague.

[104] What memory gives me on this subject is not condensed into a tangible, precise fact that I could relate here in a more or less circumstantial way. Here again memory is limited to a kind of fog, which nevertheless gives me an overall impression that I must try to pin down. It's the impression left on me by a certain inner attitude, which must have ended up becoming like second nature, and which manifested itself every time I received mathematical information about something that was more or less ‘up my street’. To tell the truth, in some relatively harmless way, this attitude must have always been mine, it's part of a certain temperament, and I've had the opportunity to touch on it in passing. It's about this reflex of first agreeing to read only a statement, never its demonstration, to try first of all to situate it in what is known to me, and to see if in terms of what is known the statement becomes transparent, obvious. This often leads me to reformulate the statement in a more or less profound way, in the sense of greater generality or greater precision, often both at the same time. It's only when I can't manage to ‘fit’ the statement in terms of my experience and my images that I'm prepared (almost unwillingly at times!) to listen to (or read...) the ins and outs that sometimes give ‘the’ reason for the thing, or at least a demonstration, understood or not.

This is a peculiarity of my approach to mathematics which, it seems to me, set me apart from all the other members of Bourbaki when I was part of the group, and which made it virtually impossible for me to take part in collective work as they did. This peculiarity must also have been a handicap in my teaching activity, a handicap that must have been felt by all my students until today when (with the help of age) it has finally softened somewhat.

This trait in me is surely already indicative of a lack of openness. It implies only partial openness, ready to accept only what ‘comes at the right time’, or at least very reluctant to accept anything else. In the choice of my mathematical investments, and the time I agree to devote to this or that unexpected information, this deliberate choice of ‘partial closure’ is today stronger than ever. It is even a necessity, if I want to be able to follow the call of what fascinates me most [◊ 105], without still giving ‘my life to devour’ to lady mathematics!

The ‘fog’, however, gives me back more than just this particularity, which I came to realise a few years ago (better late than never!). At a certain point, this reflex became like a point of honour; it would be the devil if I didn't manage to ‘get’ this statement (assuming it wasn't already quite familiar to me) in less time than it takes to say it! If it was an illustrious stranger who had made the statement, there'd be the added nuance that I (who'm supposed to be in the know, after all!) wouldn't already have all that up my sleeves! And very often I did have it, and beyond that - my attitude would have tended to go something like this: ‘OK, you can go and get dressed - you'll come back when you've done a bit better!

That was precisely my attitude in the case of the ‘young white boy who stepped on my toes’. I couldn't even swear that there weren't interesting details in what he was doing that weren't covered by what I'd done in my ‘secret notes’ - which is incidental, by the way 2 . Finally, this episode also sheds light on the question I am examining here: that of a profound disturbance in this openness to the beauty of mathematical things. It was as if, from the moment I had ‘done’ such and such a thing, its beauty had disappeared for me, and all that remained was a vanity that claimed credit and profit. (Although I didn't deign to take the time to publish it - it's true that there would have been too much of that). It was a typical attitude of possession, analogous to that of a man who, having known a woman, no longer feels her beauty and runs after a hundred others without suffering for all that that another knows her. It was an attitude that I disapproved of in love life, believing myself to be far above such vanity, while being careful not to notice the obvious fact that it was indeed my attitude towards mathematics!

I have a sort of impression that these crude competitive dispositions, ‘sporting’ dispositions if you like, which I have just put my finger on in my person, must have started to become common in ‘my’ mathematical milieu, around the time they were common in me. I would be hard pressed to place in time the moment when they appeared, or when they became like an intimate part of the air we breathed in this [◊ 106] milieu or the air my students breathed when they came into contact with me. The only thing I think I can say is that it must have taken place in the 1960s, perhaps as early as the early 1960s, or the late 1950s. (If so, all my pupils were entitled to it - it was take it or leave it for them!) To be able to place it, I would need other specific cases, which at the moment totally escape my memory.

This humble reality was, of course, in complete contrast to the noble image I had of my relationship with mathematics, and with young researchers in general. The crude subterfuge I used to fool myself was meritocratic in inspiration: as far as this image was concerned, all I retained was the relationship with my students (who contributed to my prestige, of which they were the noblest jewels! ), and to the particularly brilliant young mathematicians whose merits I had been able to recognise and whom I treated on an equal footing, just like my students, without waiting for their heads to be crowned with laurels (which of course didn't take long - you either have ‘flair’ or you don't!). As for the young people who didn't happen to be among my pupils, or among those of one of my friends, or to be young geniuses, I wasn't at all concerned about my relationship with them. They didn't matter.

I think that this reality was usually softened, tempered, when I found myself in a personal relationship with the young researcher, either because I met him at my seminar or because he had written to me. It may be that the case of the ‘young white boy’ is, from this point of view, something of an exception. It seems to me that in the case of the researchers I've just mentioned, I must have considered them to be ‘under my protection’, and that must have awakened in me a more benevolent attitude. In this case too, my desire to put myself forward could find an outlet, by giving my comments to the person concerned and making suggestions about how to take up his work in a broader perspective, perhaps, or by getting to the bottom of things. In such a case, there's a good chance that the young researcher, who for a limited time took on the role of a pupil, would also find this to his advantage, and that he would have fond memories of his relationship with me. (Any feedback on this subject would be most welcome).

I was thinking here mainly of the case of younger researchers, whereas [◊ 107] the ‘sporting’ attitude was by no means limited to my relationship with them, needless to say. But it is in the relationship with young researchers, surely, that both the psychological and practical impact of a mathematician in the limelight tends to be strongest, most fraught with consequences for their future professional lives.

​ (41) Off the merry-go-round!

I stopped last night with a feeling of relief, of great satisfaction, the contentment of someone who hasn't wasted his time! I suddenly felt light and joyful - a joy that was a little mischievous at times, bursting into mischievous laughter - the laughter of a joking rascal. All I'd done was watch an episode I'd already ‘seen’, the famous ‘white boy who...’, from a slightly different angle. An angle showing my relationship with mathematics itself, in certain circumstances, not just my relationship with mathematicians. That was all it took for a cherished myth to go up in smoke.

To tell the truth, this wasn't the first time I'd looked at my relationship with mathematics. Two and a half years ago I had already spent a few weeks or months on it. At that time I realised (among other things) the importance of egotistical forces, forces of self-aggrandisement, in my past investment in maths. But last night I had just put my finger on an aspect that had escaped me at the time. Now that I'm coming back to it, I realise that this aspect, the aspect of the jealous attitude in my relationship with maths, ties in with the ‘simple’ discovery that came at the end of the first night when I ‘meditated’ (meditating without knowing it, like Mr Jourdain writing prose...). It's quite possible that this had something to do with the joyful exultation that followed. Even if it wasn't consciously perceived, it was a bit like the reconfirmation, in a new light, of something I'd discovered earlier - and the pleasure then is the same as in mathematics, when without having looked for it you come across, by an entirely different route, something you know, something you've found perhaps years before. Each time this is accompanied by a feeling of intimate satisfaction, as the harmony of things is revealed once again, and at the same time our knowledge of them is more or less renewed.

[108] What's more, I think this time I've really ‘come full circle’! For days I'd had a strong feeling that there was still something to bring to light, without being able to say very clearly what. I didn't try to force it, I felt I just had to let it happen, letting the thread I was following unfold freely, through landscapes that were both familiar and unexpected. Unexpected, because until now I'd never bothered to look at them. It was at a walking pace that I approached the remaining ‘hot spot’. And I think it's the last one on the journey I've just made, which is coming to an end.

And as soon as I reached this point, I had the impression of someone arriving at a belvedere, from which they can see the unfolding landscape they have just covered, of which at any given moment they could still only perceive a portion. And now there is this perception of expanse and space, which is a liberation.

If I try to put into words what the landscape in front of me is giving me, I come up with this: everything that has come to me, and often unwelcome and unwelcome, in my life as a mathematician in recent years, is the harvest and message of what I sowed, when I was part of the world of mathematicians.

Of course, I've said this to myself over and over again over the years, and even in the notes I've just written. I've said it to myself by analogy, to some extent, with other harvests that have come to me insistently, that I've rejected for a long time and that I've ended up welcoming and making my own. From the first one I accepted, even before I knew anything about meditation, I understood that every harvest had to have a meaning, and that to be reluctant was only to evade a meaning and put off the deadline for a conclusion. This knowledge has been invaluable to me, because it has often kept me from feeling pity for myself, and from the righteous indignation that is often a disguised form of it. This knowledge is in me like a half-maturity, which does not yet put an end to the inveterate reflex of refusing the harvest when it seems bitter. When I say to myself ‘there's no point in begrudging’, that doesn't mean that the harvest is welcomed. I don't pity myself, or perhaps feel indignant, and yet I ‘balk’! As long as the food isn't eaten, it isn't welcomed - and not to eat is to begrudge it.

Welcoming and eating is work: a certain amount of energy [◊ 109] ‘works’, work is done in broad daylight or in the shadows, something is transformed... Whereas to balk is to waste energy that is dispersed - to ‘balk’! And we can't do without the work of eating, digesting and assimilating. The mere fact of going through events, of ‘doing’ or ‘acquiring’ an experience, has nothing in common with work. It's simply possible material for work that you are free to do, or not to do. In the thirty-six years since I first encountered the world of mathematicians, I have made use of this freedom that I have, by avoiding a job, while the material, the substance to be eaten and digested, increased from year to year. This feeling of joyful liberation that I've been experiencing since yesterday is surely a sign that the work that was in front of me, that I kept putting off in favour of other work or tasks, has finally been done. It's about time indeed!

It's still too early to be sure that this is indeed the case, that there isn't still some obscure and stubborn corner that has escaped my attention and that I'll have to come back to. But it's also true that this feeling of liberation is not misleading - every time I've felt it in my life, I've been able to see afterwards that it was indeed a sign of liberation; of something lasting, something acquired, the fruit of an understanding, a knowledge that has become part of myself. I am free, if I wish, to ignore this knowledge, to bury it where I wish and how I wish. But it is not in my power or anyone else's to destroy it, any more than you can destroy the ripeness of a fruit, make it return to a state of greenness that is no longer its own.

It's a great relief to have it confirmed once again that I'm not ‘better’ than the others. Of course, that too is something I repeat to myself quite often - but repeating and seeing are not the same thing! Lacking the innocence and mobility of a child, who sees as he breathes, it often takes work to see the obvious - and now I've finally seen it: I'm not ‘better’ than some colleagues or ex-students who, just a few days ago, ‘took my breath away’! Just think of the weight I've been relieved of! It may be gratifying in a way to think you're better than the others, but it's also very tiring. It's an extraordinary waste of energy even - as it is every time you have to maintain a fiction. We rarely realise it [◊ 110], but it already takes a lot of energy, just to maintain the fiction against all odds, while the evidence at every step proclaims in my carefully plugged ears that it's all bogus, just look at it, you idiot! It may be a job sometimes to see, but when it's done, it's done. It saves me, once and for all, having to go around plugging my eyes and ears all the time - that's got to be done too! and having to suffer an intolerable outrage every time something falls on me that I've inadvertently put there.

I've had enough of this merry-go-round! Once you've seen the merry-go-round, you're already off it. You've paid for it, all right, I've got the right to go for life on it, and even the duty to do so, as everyone will tell me: right, duty - it's up to the customer. It's very tiring too, all these rights that are duties and all these duties that are rights, which stick to me when I think I'm better than the others. It's normal after all, when you're better, you cash in discreetly (that's ‘rights’) and you ‘pay’, you do your duty for the honour of the human spirit and mathematics - it's very beautiful, it's true, honour, spirit, mathematics, who could say better, bravo! bis! It's all very beautiful, but it's also very tiring, it ends up giving you a stiff neck. I've had my torticollis and now that's enough - I leave room for others to stand stiffly.

It's also normal (since I was talking about pupils) for the pupil to overtake the teacher. I took offence at that, I had energy to waste! But no more!

What a relief!

  1. (August 8) After checking, it appears that the beginnings of my reflection on motives are at the beginning, not at the end of the 1960s.
  2. (August 8) It has since occurred to me that this thing is not so ’incidental‘ as all that, that it constitutes the line of passage from the ’sporting attitude‘ to the beginning of dishonesty, a line that I may have crossed from time to time...

VII THE CHILD HAS FUN

Contents

  1. The Child
  2. The troublemaker boss - or the pressure cooker
  3. Turning the tables
  4. The Guru-not-Guru - or the three-legged horse

​ (42) The Child

It's even certain that there must be nooks and crannies where the broom hasn't been. It doesn't matter, they'll come to my attention and then there'll always be time to take care of them. But as far as my famous ‘mathematical past’ is concerned, the big clean-up is done, no doubt about it.

Now that I've seen once again that I'm no better than anyone else, I mustn't fall back into the same old trap of thinking I'm better than myself! To think I'm better now, off the merry-go-round and all that, than I was fifteen years ago, or a fortnight ago. I've learnt something [◊ 111] during these fifteen years, that's for sure, and during the fortnight too and even since yesterday. When I learn something I mature, I'm not quite the same. I'm not ‘better’ when I've learnt something than when I was still learning it. A riper fruit is not ‘better’ than a less ripe or green fruit. One season is not ‘better’ than the one before it. The taste of the ripest fruit may be more pleasant, or less pleasant, depending on taste. I feel better about myself from one year to the next, so I guess the changes I'm going through are ‘to my liking’ - but they're not to the liking of all my friends and family. Every time I go back to doing maths, I get compliments from all sides, like: ‘What an idea he had to do something else! Everything's back to normal, it's about time! It's worrying to see someone change...

I'm learning, I'm maturing, I'm changing - to the point where sometimes I find it hard to recognise myself in the person I was and now rediscover, through a memory or the unexpected testimony of others. I change, and there's also something that remains ‘the same’. It's always been there, probably since I was born, and perhaps even before. I seem to have been able to recognise it quite well in recent years. I call it ‘the child’. Because of this, I'm no better at this moment than at any other time in my life; he was there, even if it would often have been difficult to guess his presence. With this thing too, I am no better than anyone else, and no one is better than me. At certain moments or in certain people, the child is more present. And that's a very good thing. It doesn't mean that someone is ‘better’ than someone else, or than themselves at another time.

Often, when I'm doing maths, or making love, or meditating, it's the child who's playing. He's not always the only one ‘playing’. But when he's not there, there's no maths, no love, no meditation. There's no point pretending - and I've rarely acted that way.
It's not just the child, that's for sure. There's the ‘me’, the ‘boss’ or the ‘big boss’, call him what you will. Surely the boss is indispensable to the running of the company. If there's a boss, it must be for something. He's in charge of everything, and like all bosses, he has an annoying tendency to become intrusive. He takes himself terribly seriously [◊ 112] and in the end wants to be better than the boss opposite. Invasive or not, he's only the boss, he's not the worker. He organises, he orders, and he cashes in, that's for sure! - He takes the profits as his due, and suffers the losses as an outrage. But he creates nothing. Only the worker has the power to create, and the worker is none other than the child.

It's rare to find a company where the boss and the worker get along. More often than not, the worker is nowhere to be seen, locked away in God knows where. It's the boss who pretends to take his place in the workshop, with the results you can guess. And often, when the worker is actually there, the boss wages war on him, either violently or through skirmishes - not much comes out of the workshop! Sometimes, too, the boss is suspiciously tolerant of the worker, and lets him get on with it, grumbling, without taking his eyes off him. It's like a constantly renewed truce in a war that has never stopped. And the worker is able to get some work done thanks to the truce.

It's not at all certain that, by virtue of the meditation I've just made, the attitude of possessiveness in me towards mathematics has disappeared as if by magic! At the very least, I'd have to take a much closer look at the manifestations of possessiveness, one of which I've just touched on by name. This is not the place for this ‘introduction’, which has become an ‘introductory chapter’, which in turn is already getting long! One thing did ‘click’ last night, though, and I'd like to come back to it now, something I'd noted with some surprise two or three years ago.

I was working on a mathematical question, I don't know what it was, and at some point (through some circumstance) it occurred to me that the question I was looking at had perhaps already been looked at, that it might well be dealt with in black and white in some book that it was up to me to consult in the library. The mere mention of this possibility had a lightning effect that stunned me: from one moment to the next, the desire had disappeared. Suddenly, the question I'd perhaps spent weeks thinking about, and was prepared to spend weeks more, had lost all interest for me! It wasn't spite, it was a sudden and total lack of interest. If I'd had the book in my hands, I wouldn't have bothered to open it.

In fact, the possibility wasn't confirmed, so the desire [◊ 113] returned and I carried on as if nothing had happened. I was still taken aback. Of course, if I'd really needed what I was doing to do something else, there wouldn't have been such a spectacular drop in interest. I've often gone back to familiar things, knowing or suspecting that they were familiar without the slightest concern. At the time I was on a path where it was more economical, and above all much more interesting, to do things my own way, as they presented themselves to me, than to go digging through books or articles. I'd do it ‘on the run’ to something else, where my desire was leading me. And of course, I was ‘in the know’ enough to know that what was at the end was not in any book or article.

This reminded me that mathematical work, even if done alone for years on end, is not purely personal, individual work, like meditation is - at least not for me. The ‘unknown’ that I pursue in mathematics, for it to attract me with such force, must not only be unknown to me, but unknown to everyone. What is written in mathematical books is not unknown, even if I myself have never heard of it. Reading a book or an article has never attracted me; I have avoided it whenever I could. What it can tell me is never unknown, and the interest I take in it does not have the quality of desire. It's a circumstantial ‘interest’, an interest in information that can be useful to me, as an instrument of a desire of which it is in no way the object.

On reflection, it doesn't seem to me that the event I reported is the sign of a jealous, possessive disposition, the sign of a vanity that has been disappointed. There was no spite or disappointment in me, just the sudden disappearance of a desire that, just a moment before, had been intense. This was at a time when I had absolutely no thought of publishing anything, nor of one day taking it into my head to publish something again. This desire was not an expression of vanity, of the craving to accumulate knowledge, titles and credits - it was a real desire, the desire of a child who is passionate about play. And suddenly - nothing! I don't understand... Sorry!

​ (43) The troublemaker boss - or the pressure cooker

[◊ 114] I feel that I have finally completed this retrospective of my life as a mathematician. Of course, I have not exhausted my subject - it would take volumes, assuming that such a subject could be ‘exhausted’. That wasn't my point. My point was to get to the bottom of whether or not I had been a stakeholder and co-actor in the appearance of a certain ‘air’ that I now feel in puffs, and if so, in what way. Now I know for sure, and it feels good. It could be fascinating to go further, to delve deeper into what has only been glimpsed or touched upon. There are so many exciting things to see, do and discover! As far as my past as a mathematician is concerned, it seems to me that what I needed to look at in order to take on that past has been seen.

Undoubtedly, if I went deeper into this meditation, I would not fail to learn many interesting things about my present. One thing that this work has made me feel almost every step of the way is the extent to which I have remained attached to this past, the importance it has had to this day in my self-image, and also in my relationship with others; especially in my relationship with those I have, in a certain sense, left behind. My relationship with this past has undoubtedly changed in the course of this work, in the direction of detachment, or a greater lightness. The future will tell me more. But it is likely that an attachment will remain, as long as my mathematical passion is not burnt out and quenched - as long as I ‘do maths’. And I don't care to guess or predict whether it will die before I do...

For more than ten years I had thought this passion had died out. It would be truer to say that I had decreed that it was extinct. That was the day I stopped doing maths for a while and rediscovered the world! For three or four years I was absorbed in an activity so intense that my old passion couldn't have found the slightest gap through which to slip and manifest itself. Those were years of intense learning, at a certain level that remained fairly superficial. In the years that followed, the mathematical passion manifested itself in sudden, totally unexpected bursts. These would last a few weeks or months, and I would persist in ignoring their fairly clear meaning. I'd decided once and for all that the urge to do maths, which was decidedly useless, was now a thing of the past, full stop! But the ‘good-for-nothing’ didn't hear it that way - and I remained deaf.

[◊ 115] Paradoxically, it was after I discovered meditation (in 1976), with the arrival of a new passion in my life, that the reappearances of the old one were particularly strong, almost violent - as if each time a lid popped off under too much pressure. It was only five years later, under the pressure of events, so to speak, that I took the trouble to examine what was happening. It was the longest meditation I'd ever done on a seemingly well-defined issue: it took me six months of stubborn and intense work to get round a kind of iceberg, the visible tip of which had ended up becoming embarrassing enough to force me, almost unwillingly, to look into it. There was a situation of conflict, which appeared to be the conflict of two forces or desires: the desire to meditate and the desire to do maths.

In the course of this long meditation, I learnt step by step that the desire to do maths, which I treated with disdain, was, just like the desire to meditate, which I valued highly, a child's desire. The child has nothing to do with the disdain or modest pride of the big boss! The child's desires follow one another, as the hours and days go by, like the movements of a dance arising from one another. That is their nature. They are no more opposed than the stanzas of a song, or the successive movements of a cantata or a fugue. It is the bad conductor who declares that one movement is ‘good’ and another ‘bad’, and who creates conflict where there is harmony.

After this meditation, the boss has calmed down and is less inclined to stick his nose where it doesn't belong. The work took a long time this time, whereas I thought it would be done in a few days. Once the work is done, the ‘result’ seems obvious, and can be expressed in a few words (37). But if someone perceptive had said these words to me before or during the work, it probably wouldn't have done me any good. If the work took so long, it's because the resistance was strong and deep-rooted. The boss took it in his stride, and he never batted an eyelid, because it was happening in an atmosphere where there was no way he'd get angry. One thing's for sure, it's been six months well spent, and I couldn't have done without them; any more than a woman can do without the nine months of pregnancy to finally give birth to something as ‘obvious’ as a brat.

​ (44) Turning the tables

[116] I haven't meditated for a year and a half now, apart from a few hours in December, to get to the bottom of an urgent question. And it's been a year since I put most of my energy into doing maths. This ‘wave’ came like the others, maths-waves or meditation-waves: they come without announcing their arrival. Or if they do, I never hear them! The boss has a slight preference for meditation, it seems: each time the meditation wave is already followed by a maths wave, whereas I thought it would last forever; and the maths wave, which (it seemed to me) was a matter of a few days or weeks at most, lingers and extends over months and perhaps even, who knows, years. But the boss has come to understand that he's not the one who makes these rhythms and that he has nothing to gain by trying to regulate them.

But perhaps there was finally a change in the boss's ‘little preference’, because it's been almost a year since it was agreed and decided, and I'm off to ‘do maths again’ for a few years at least, officially so to speak: I've even applied for a job at the CNRS! More importantly, and completely unexpected even a year ago, I'm starting to publish again. Even after the 1981 meditation I mentioned earlier, when the desire to do maths ceased to be treated as a poor relation, the idea would never have occurred to me that I might start publishing maths again. Something else at a pinch, a book where I'd talk about meditation, or dreams and the Dreamer - and even then, I was far too busy with what I was doing to want to write a book about it! And for what?!
So it was a rather important decision, one that would affect the course of my life for years to come, and one that was taken somewhat on the spur of the moment, I'm not even sure when and how. One day, when there started to be a whole bunch of typed notes (well, well! until then I'd just handwritten my mathematical cogitations... (38)), on homotopic fields and models, etc., it was decided: we're publishing this! And while we're at it, we might as well pull out all the stops and start a little series of mathematical reflections, the name of which was obvious - all we had to do was capitalise it: Réflexions mathématiques! That's more or less what this famous ‘fog’, which so often takes the place of a memory, is telling me at the moment. A memory that is surely very shortened, in this case [◊ 117]. The remarkable thing, in any case, is that this thing was done without even pausing to look at where I was going, what was pushing me, or carrying me... That's what I would still like to do, on the momentum of this unforeseen meditation, to be able to feel it as truly completed.

The question that immediately springs to mind is: is this ‘remarkable thing’ that I've just noticed a sign of the (so-called?) ‘discretion’ of the boss, who wouldn't want to interfere (even with an indiscreet glance...) in such a beautiful spontaneous movement that doesn't need him at all, etc.; or is it a sign, on the contrary, that he's taken sides outright, and that the so-called ‘small preference’ is pushing him all the way in the direction of maths?

It was enough to put the question in black and white for the answer to appear! It wasn't the kid, who had embarked on a longer game than others, perhaps, who decided that he was going to carry on for X number of years without a hitch, and wisely fill in as many pages as it took to make a reasonable number of volumes of a fine series with capital letters! The boss has planned and organised everything, and all the kid has to do is get on with it. Maybe the kid won't ask for anything better, there's no way of knowing in advance - but that's a secondary question. What the kid wants depends, to a certain extent at least, on the circumstances, which depend above all on the boss.

The boss has clearly made up his mind. In fact, he has just shown a certain flexibility, since a meditation has been going on for over a month under his benevolent eye. It's also true that his benevolence is by no means disinterested, since the tangible product of the meditation, the notes I'm in the process of writing, will be the most beautiful cornerstone of the tower he already sees himself building, with the stones graciously cut by the child-labourer, who is apparently well-disposed. Decidedly, it's a bit early to be complimenting him on his ‘flexibility’! A few hours of meditation three months ago, and all in all a year and a half from now, would seem rather meagre indeed!

However, I don't have the impression that during all that time [◊ 118] there was a desire for meditation that had been repressed or frustrated. In the few hours in December, I took stock and saw what I had to see; that was enough to transform a situation, which had not been clear. I resumed the thread of the interrupted mathematical work, without having to cut short anything else. It doesn't seem to me that a conflict has reappeared in the background, I mean the one that was resolved more than two years ago and which would have reappeared this time in reverse form. It's in the boss's nature to have preferences, and that's his right - it would be silly for him to pretend not to (although more silly things happen than that...). This is not a sign of conflict, even if it is often the cause of it. As things stand, it really doesn't seem that he's to blame for a lack of flexibility!

With that out of the way, it remains for me to try and pinpoint the boss's ‘motivations’ for this turnaround, which has taken place as discreetly as possible, and yet which, on closer inspection, is quite spectacular.

​ (45) The Guru-not-Guru - or the three-legged horse

This immediately brings me back to the meditation that I continued from July to December 1981, after a period of four months that I had just spent in a kind of mathematical frenzy. This somewhat insane period (which, incidentally, was very fruitful from a mathematical point of view (39)) had ended, overnight, following a dream. It was a dream that described, in a parable of irresistible savage force, what was happening in my life - a parable of this frenzy. The message was dazzlingly clear, yet it took me two days of intense work to accept its obvious meaning (40). Once I had done that, I knew what I had to do. I didn't come back to this dream in the course of my work for the next six months, but all I was doing was penetrating further into its meaning and fully assimilating its message. The day after the dream, this message was understood on a level that remained superficial and crude. What I needed to go deeper into, above all, was ‘my’ relationship; that of the boss, I mean, to the two desires involved, which appeared to me to be antagonistic.

So much has happened in my life since that meditation that it seems to me to be in the very distant past. If I try to [◊ 119] formulate what I have retained from what it taught me about the motivations of the ‘boss’, it comes to this: during the twelve years that had then elapsed since the ‘first awakening’ (of 1970), the boss had bet on what was obviously ‘the wrong horse’: between mathematics and meditation (which he liked to set one against the other), he had opted for meditation.

That's one way of putting it, since the thing and the name ‘meditation’ hadn't entered my life until October 1976, five years earlier. But in my beloved image of myself, which had been given a new coat of paint in 1970, meditation had come at just the right moment, six years later, to add lustre to a certain attitude or pose that I had spotted for a long time but never examined until this meditation in 1981. I called it the ‘master's syndrome’, and some have also called it (quite rightly) my ‘Guru pose’. If I adopted the first designation rather than the second, it was undoubtedly because it encouraged confusion about the nature of the thing, which I liked to maintain. From my earliest childhood there was a spontaneous pleasure in teaching which was in no way opposed to the spontaneous pleasure in learning, and which had nothing of the nature of a pose. It was this force above all that was at play in my relationship with my students; this relationship was superficial, but it was strong and good-natured, by which I mean: without pose. It was after what I call my ‘awakening’ in 1970, when a world that had been familiar to me was retreating to the point of almost disappearing, and with it the pupils and the opportunities I had to ‘teach’, to share things I knew and which for me had meaning and value - that's when ‘the boss’ took his revenge as best he could: Instead of teaching maths, which was just good enough to earn a living, but otherwise unworthy of my new greatness, I saw myself teaching a certain ‘wisdom’ by my life and example. Of course, I was careful not to say anything of the sort either to myself or to others, and when I received echoes in this direction, I was sure to have to recuse myself, saddened by so much incomprehension on the part of such friends or relatives. No matter how many times I explained it to them, they still didn't understand, which was a sorry state of affairs for them!

I had read a book or two by Krishnamurti which had made a strong impression on me, and in no time at all my head had assimilated a certain message and [◊ 120] certain values (41). That was all it took to believe that everything had happened (while pretending otherwise, of course). I didn't need to read any more, I was capable of improvising the purest Krishnamurti in speech and writing, in a speech of flawless coherence. But no matter how beautiful and flawless the discourse, at no time did it seem to be of any use to me or to anyone else. This went on for years without me even pretending to learn anything from it. When I discovered meditation, the jargon left me overnight, without a trace. That's when I realised the difference between a speech and knowledge.

The big boss rectified the situation immediately: Krishnamurti out, meditation in! Discreetly, it goes without saying, he now had to play with a completely different baton. Times had changed, with this kid who now ran between his legs, and who was a bit sharp-eyed at times. I guess the kid was busy elsewhere. In any case, it was only five years later, when a certain cauldron had exploded and the kid had run to see what was going on, that the great chef's scheme was uncovered.
After all, it wasn't that long ago, just over two years ago, that the Guru-without-a-hair was finally exposed - just another disguise down the drain! The poor boss was about to be stripped naked. Or to put it another way: the ‘Meditation’ horse, which had taken the place of the horse with no name (which certainly shouldn't have been called ‘Krishnamurtian’!) was making really derisory returns on his bets, especially if you compare them with the handsome returns of the ‘Mathematics’ horse in the old days when the boss was still betting on him. If he maintained the wrong bet for so long, it was out of sheer inertia - he had already changed his bet once, which isn't that common, and it took all the impact of a striking event to do so (42). Bosses don't really like to change their stakes - and in this case it was a sort of going back to the previous stake.

It was from 1973 onwards, when I retired to the countryside, that the returns from the new horse began to be really meagre compared with those of yesteryear. The unexpected appearance of meditation three years later gave them a bit of a boost. There was even a dizzying episode from March to July 1979, which I won't go into here, when I once again [◊ 121] took on the role of apostle, this time apostle of a wisdom that was both immemorial and new, sung in a poetic work of my own composition that I finally refrained from entrusting to a publisher (43). But two years later, with the Guru definitively out of commission, it was as if the Meditation horse had broken a leg (as far as returns to the boss were concerned) - there was no longer any way, fingering or no fingering, of playing the gurus!

After that, things didn't hang on much longer - the three-legged horse out the window, along with the Apostle-Poet, The Gurupas-Guru and Krishnamurti-who-dare-not-say-his-name. And long live mathematics!

We look forward to what happens next...

VIII THE SOLITARY ADVENTURE

Contents

  1. The forbidden fruit
  2. The solitary adventure
  3. Giving and receiving
  4. Observation of a division
  5. The weight of a past

​ (46) The forbidden fruit

I had to stop writing for two days. After careful re-reading, it seems to me that the above scenario is indeed, roughly speaking (grosso modo — translator's note), a description of reality, a description that now needs to be explored a little further. In particular, I'd have to take a closer look at the respective merits of the two ‘horses’, meditation and mathematics; and also try to understand which events or circumstances ended up triggering the ‘switch’ in the boss's bet, against the forces of inertia that would have led him to keep his losing bet indefinitely.

Perhaps we also need to sound out the kid's preferences. It's common knowledge by now that he wants to change his game from time to time, and apparently the boss is flexible enough not to force him at all costs to always play this game and never that. Over the last few years he has learnt to take the kid into account, to work with him, without waiting for the kettles to explode. It's not complete harmony, but it's no longer war, more a kind of entente cordiale, which occasional tensions tend to soften rather than harden.

When he's not opposed too harshly, the kid is by nature quite flexible in his preferences. (Unlike the boss, who eventually learnt a modicum of flexibility only unwillingly and in his old age...) But the fact that the kid is flexible doesn't mean that he doesn't have his own preferences, that he isn't more strongly attracted to one thing than another.

It's not always easy to see clearly, to distinguish between [◊ 122] the child's desires and the boss's preferences, or even what the boss has decided once and for all. When I said to myself a while ago: meditation is better, more important, more serious and all that than mathematics, for such and such reasons (of the most pertinent, no doubt), it was the boss who gave himself good reasons afterwards to convince himself that the bet he was making was indeed ‘the right one’. Kids don't say that one thing is ‘better’ or ‘more important’ than another. He's not one for speeches. When he feels like doing something, he just goes ahead and does it if no one is in his way, without questioning whether it's ‘important’ or ‘better’. Their desires vary from one thing to another and from one moment to another. To detect his preferences, there's no point in listening to the boss's explanatory speeches, when he claims to speak for the child when he can only speak for himself. It's only by observing the child at play that we can perhaps detect his predilections. And even then it's not so obvious: when he plays this with gusto, it doesn't always mean that he wouldn't play something else with delight, if the boss gave him a helping hand.

Visibly, what attracts him more than anything else is the unknown - it's pursuing the nebulous recesses of the night and bringing into the light of day what is unknown to him and to everyone else. And I have the impression that when I added ‘and everyone else’, I was referring to the child's desire, and not to the boss's vanity, who wants to impress the gallery and himself. It's also a well-known fact that what the kid brings back every time from the shadows of inexhaustible attics and cellars are ‘obvious’, childish things. The more obvious they seem, the happier he is. If they're not, it's because he hasn't done his job to the end, because he's stopped halfway between darkness and daylight.

In maths, the ‘obvious’ things are also those that sooner or later someone must stumble across. They are not ‘inventions’ that you can do or not do. They're things that have always been there, that everyone comes into contact with without paying any attention to them, even if it means making a long diversions around them or tripping over them every time. After a year or a thousand, infallibly, someone ends up paying attention to the thing, digging around it, digging it up, looking at it from all sides, cleaning it up, and finally [◊ 123] giving it a name. This kind of work, my work of predilection, another every time could do it, and what is more, another could not fail to do it one day or another (44).

It's not at all the same when it comes to discovering myself, in the not at all collective game of ‘meditation’. What I discover, no other person in the world, today or at any other time, can discover for me. It is up to me alone to discover it, that is to say, also: to assume it. This unknown is not destined to be known, almost by force of circumstance, whether or not I take the trouble to be interested in it. If it waits in silence for the moment when it will be known, and if sometimes, when the time is ripe, I hear it calling, it is only I, the child in me, who is called to know it. He is not a stranger on borrowed time. Of course, I am free to follow his call, or to evade it, to say ‘tomorrow’ or ‘one day’. But the call is addressed to me and to no one else, and no one but me can hear it, no one else can follow it.

Every time I've followed this call, something has changed in ‘the enterprise’, more or less. The effect has been immediate, and immediately felt as a blessing - sometimes as a sudden release, an immense relief from a weight that I often carried without even realising it, and whose reality manifests itself in this relief, this liberation. On a lesser scale, such experiences are common in any work of discovery, and I've had occasion to talk about them. However, the thing that distinguishes the work of self-discovery (whether it takes place in the open or remains underground) from any other work of discovery is precisely that it really changes something in the ‘business’ itself. It's not a quantitative change, an increase in output, or a difference in the size or even the quality of the products leaving the workshop. It's a change in the relationship between the boss and the worker-child. Maybe there's even a change in the boss himself, if that means something other than his relationship with the worker-child. For example, he may look less at production - but this is also an aspect of his relationship with the worker, through the emergence of a concern or respect that was perhaps foreign to him before. In all the cases where I meditated, the change was in the [◊ 124] direction of a clarification and appeasement in the relationship between boss and worker. Except in certain cases where the meditation remained superficial, meditations ‘of circumstance’ under the sole pressure of an immediate and limited need, the clarification has lasted until today, and so has the appeasement.

This gives the work of self-discovery a different sense from any other work of discovery, even though many essential aspects are the same. There is a dimension to self-knowledge, and to the work of self-discovery, that sets it apart from all other knowledge and all other work. Perhaps this is the ‘forbidden fruit’ of the Tree of Knowledge. Perhaps the fascination that meditation has exerted on me, or rather the fascination of the mysteries whose existence it has revealed to me, is the fascination of the forbidden fruit. I have crossed a threshold where fear has disappeared. The only obstacle to knowledge is inertia, sometimes considerable inertia, but it is finite and by no means insurmountable. I felt this inertia at almost every step, insidious and omnipresent. It sometimes exasperated me, but never discouraged me (no more than in mathematical work, where it is also the main obstacle, but of incomparably lesser weight). This inertia becomes one of the essential ingredients of the game; one of the protagonists, to put it better, in this delicate and by no means symmetrical game that has two of them - or three, to put it better: on the one hand the child who rushes forward, and the boss (who is inertia) who puts the brakes on everything he can (while pretending not to be there), and on the other the glimpsed form of the beautiful stranger, rich in mystery, both near and far, who both evades and calls...

​ (47) The solitary adventure

This fascination with ‘meditation’ has been of considerable power for me - as powerful as the attraction of ‘woman’, whose place it seems to have taken. The fact that I have just written ‘has been’ does not mean that this fascination has now been extinguished. In the year that I've been involved in mathematics, it has only faded into the background. Experience tells me that this situation can be reversed overnight, just as this situation is itself the effect of an entirely unforeseen reversal. In fact, during each of the four long periods of meditation that I went through (one of which lasted for almost a year and a half), it was something that I took for granted that I was going to keep going until my [◊ 125] last breath, to probe as far as I could into the mysteries of life and those of human existence. When the notes piled up in impressive stacks to the point of threatening to overwhelm my workroom, I even ended up having a piece of furniture made to measure to accommodate them, with plenty of room (by a quick calculation of arithmetic progression) to accommodate those that would soon be added over the years; I had allowed for a margin of about fifteen years if I remember correctly (which was already starting to happen!). In this case the boss had done things right, for stewardship it was great stewardship! That, and a large-scale tidying up of all the personal papers closely or remotely linked to the meditation work, was in fact the last task he undertook and brought (almost) to a successful conclusion, just before the switch of preferences and bets. One wonders whether he did not have an ulterior motive, and whether he did not already see tomes of Mathematical Reflections filling the empty shelves supposedly intended for the ‘Notes’ to come.

It is true that the passion for meditation and self-discovery is vast enough to fill my life for the rest of my days. It's also true that the passion for mathematics has not been consumed, but perhaps that hunger will be satisfied in the years to come. Something inside me wants this, and feels that mathematics is a hindrance to a solitary adventure that only I can pursue. And it seems to me that this ‘something’ inside me is not the boss, nor one of the boss's desires (which, by its very nature, is divided). It seems to me that the mathematical passion still bears the mark of the boss, and in any case, that following it keeps my life moving in a closed circle; in the circle of ease, and in a movement that is that of inertia, certainly not of renewal.

I have wondered about the meaning of this stubborn persistence of mathematical passion in my life. When I follow it, it doesn't really fill my life. It gives me joy, and it gives me satisfaction, but it does not in itself bring true fulfilment. Like any purely intellectual activity, intense and long-term mathematical activity has a rather mind-numbing effect. I notice this in others, and especially in myself, every time I do it again. This activity is so fragmentary, it brings into play only such an infinitesimal part of our faculties of intuition, of sensitivity, that they become blunted [◊ 126] by not being used. For a long time I didn't realise this, and obviously most of my colleagues don't realise it any more than I did at the time. It seems to me that it's only since I've been meditating that I've become aware of this. If you pay attention, it's obvious - maths in large doses thickens. Even after the meditation two and a half years ago, when my mathematical passion was recognised as a passion indeed, as something important in my life - when I give myself over to this passion now, there's still a reserve, a reticence, it's not a total gift. I know that a so-called ‘total gift’ would in fact be a kind of abdication, it would be following inertia, it would be a flight, not a gift.

There is no such reserve in me for meditation. When I give myself to it, I give myself totally, there is no trace of division in this giving. I know that in giving myself, I am in complete harmony with myself and with the world - I am faithful to my nature, ‘I am the Tao’. This gift is beneficial to myself and to everyone. It opens me up to myself and to others, by lovingly untying what remains knotted within me.

Meditation opens me up to others, it has the power to untie my relationship with them, even if the other remains tied up. But it is very rare that I have the opportunity to communicate with others in any way whatsoever about the work of meditation, about this or that thing that this work has made me aware of. This is by no means because things are ‘too personal’. To take an imperfect image, I can only communicate about maths that interests me at a given moment with a mathematician who has the essential background, and who at the same time is willing to take an interest in it too. It can happen that for years I'm fascinated by such and such mathematics, without meeting (or even trying to meet) another mathematician with whom to communicate about it. But I know that if I looked for them, I would find them, and that even if I didn't find them, it would simply be a matter of luck or circumstance; that the things that interest me are bound to interest someone, or even a few people, whether ten years from now or a hundred years from now, it doesn't really matter. That's what gives meaning to my work, even if it's done in solitude. If there were no other mathematicians in the world and there should be none left, I don't think doing maths would still make sense to me [◊ 127] - and I suspect it's no different for any other mathematician, or any other ‘researcher’ of anything. This ties in with the observation made earlier, that for me ‘the mathematical unknown’ is what nobody knows yet - it is something that does not depend on my person alone, but on a collective reality. Mathematics is a collective adventure, that has been going on for thousands of years.

In the case of meditation, to communicate about it, the question of ‘baggage’ doesn't arise; not at the point I'm at, at least, and I doubt it ever will. The only question is that of an interest in the other person that responds to the interest in me. So it's a question of curiosity about what's really going on in oneself and in others, beyond the facades of rigour, which don't hide much as long as you're really interested in seeing what they cover up. But I've learnt that the moments when such interest appears in a person, the ‘moments of truth’, are rare and fleeting. It's not uncommon, of course, to meet people who are ‘interested in psychology’, as they say, who have read Freud and Jung and many others, and who want nothing more than to have ‘interesting discussions’. They have this baggage that they carry with them, more or less heavy or light, what we call a ‘culture’. It's part of the image they have of themselves, and reinforces that image, which they are careful never to examine, just like someone else who is interested in maths, flying saucers or angling. It's not this kind of ‘baggage’, nor this kind of ‘interest’, that I was talking about earlier - although the same words here designate things of a different nature.

To put it another way: meditation is a solitary adventure. Its nature is to be solitary. Not only is the work of meditation solitary - I think this is true of any work of discovery, even when it is part of a collective effort. But the knowledge that emerges from the work of meditation is ‘solitary’ knowledge, knowledge that cannot be shared, let alone ‘communicated’; or if it can be shared, it is only in rare moments. It's a work, a knowledge that goes against the grain of the most inveterate consensus, that worries each and every one of us. This knowledge is certainly expressed simply, in simple, clear words. When I express it to myself, I learn by expressing it, because the very expression [◊ 128] is part of a job, driven by intense interest. But these same simple, limpid words are powerless to communicate meaning to others, when they come up against the closed doors of indifference or fear. Even the language of dreams, which has an entirely different power and infinite resources, constantly renewed by a tireless and benevolent Dreamer, cannot get through those doors...

There is no meditation that is not solitary. If there is the shadow of a concern for anyone's approval, confirmation or encouragement, there is no work of meditation and no self-discovery. The same is true, it will be said, of any real work of discovery, at the very moment of the work itself. This is true. But outside the work itself, the approval of others, be it a close friend, a colleague, or a whole community of which we are a part, is important for the meaning of this work in the life of the person who gives himself to it. This approval and encouragement are among the most powerful incentives, ensuring that the ‘boss’ (to use that image) gives the green light without reservation for the kid to give it his all. Above all, they determine the boss's investment. It was no different in my own investment in mathematics, encouraged by the kindness, warmth and confidence of people like Cartan, Schwartz, Dieudonné, Godement, and others after them. For meditation, on the other hand, there was no such incentive. It's a passion of the young worker that the boss is basically kind enough to tolerate, more or less, because it doesn't ‘earn’ anything. It bears fruit, of course, but it's not the kind of fruit a boss wants to see. When he's not fooling himself about it, it's clear that he's not going to invest in meditation - the boss is gregarious by nature!

Only children are solitary by nature.

​ (48) Giving and receiving

As I was talking yesterday about the solitary essence of meditation, I was struck by the thought that the notes I have been writing for nearly six weeks, which have ended up becoming a kind of meditation, are nevertheless intended for publication. Inevitably, this has influenced the form of the meditation in many ways, notably through the need for brevity and discretion. One of the essential aspects of meditation, namely constant attention to what is going on inside me at the very moment of the work, only manifested itself very occasionally, and in a superficial way. Surely all this must have [◊ 129] influenced the course of the work and its quality. I feel, however, that it has the quality of meditation, above all by the nature of its fruits, by the appearance of a knowledge of myself (in this case, that of a certain past above all) that I had evaded until now. Another aspect is spontaneity, which has meant that for none of the nearly fifty ‘sections’ or ‘paragraphs’ into which the reflection has spontaneously been grouped could I have said at the outset what the substance would be; each time it was only revealed along the way, and each time the work brought to light new facts, or shed new light on facts hitherto neglected.

The most immediate sense of this work was that of a dialogue with myself, a meditation. However, the fact that this meditation is destined to be published, and moreover, to serve as an ‘overture’ to the Mathematical Reflections that are to follow, is by no means an incidental circumstance, which would have been a dead letter in the course of the work. For me, it is an essential part of the meaning of this work. If I implied yesterday that the boss must have had something to do with it (he is a master at ‘having something to do’ with everything, or very nearly so!), that in no way means that its meaning can be reduced to that - to a belated, almost posthumous ‘return’ of the famous three-legged horse! More than once I've also felt that the deeper meaning of an act sometimes goes beyond the motivations (apparent or hidden) that inspire it. And in this ‘return to mathematics’ I can see a meaning other than that of being the result-sum of certain psychic forces that were present in my person at such and such a time and for such and such a reason.

This ‘meditation’ that I am pursuing in order to offer it to those I have known and loved in the mathematical world - if I feel that it is an important part of this glimpsed meaning, it is not in the expectation that the gift will be accepted. Whether it is accepted or not does not depend on me, but only on the person to whom it is addressed. I am certainly not indifferent to whether it is accepted. But that is not my responsibility. My only responsibility is to be true to the gift I am giving, which means being myself.

What I learn from meditation are the humble and obvious things, the things that don't pay much attention. They are also the things that I won't find in any book or treatise, however learned, profound or brilliant - the things that no one else can find for me. I questioned a ‘fog’, I [◊ 130] took the trouble to listen to it, I learned a humble truth about a ‘sporting attitude’ and its obvious meaning, in my relationship to mathematics as in my relationship to others. If I'd read the Holy Scriptures, the Koran, the Upanishads - and Plato, Nietzsche, Freud and Jung on top of that, I'd be a prodigy of vast and profound erudition - all that would have done was take me away from this truth, a childish, self-evident truth. And I would have repeated Christ's words a hundred times, ‘Blessed are those who are like little children, for theirs is the kingdom of heaven’, and commented on them in fine detail, and it would only have served to keep me away from the child in me, and from the humble truths that bother me and that only a child can see. These are the things, the best I have to offer.

And I'm well aware that when such things are said and offered, in simple, clear words, they are not welcomed for all that. Welcoming is not simply receiving information, with embarrassment or even interest: ‘Wow, who would have thought...!’, or: ‘It's not so surprising after all...’. Welcoming someone often means recognising yourself in the person offering you something. It means getting to know yourself through the person of another.

​ (49) Observation of a division

This short reflection on the meaning of the present work, and on giving and receiving, comes as a digression in the thread of the reflection; or rather as an illustration of certain aspects that distinguish ‘meditation’ from any other work of discovery, and in particular from mathematical work. Yesterday I realised that these aspects have a double effect, namely two effects in opposite directions: a unique fascination for ‘the kid’, and a total lack of interest in ‘the boss’. It seems that this double effect is in the nature of things, and that it absolutely cannot be mitigated by any compromise or arrangement. Whatever you do, when the kid follows his true predilection, the boss is not happy, not at all!

There's no doubt that this is the meaning of the shift that has taken place, which could well wipe out meditation in my life in the years to come (with the exception of ‘occasional meditations’, like three months ago). I don't think these have to be entirely barren years for that, any more than last year was barren. But it's also true that [◊ 131] what I learnt there (apart from maths) is minimal, if I compare it with what I learnt in any of the four years before that. The strange thing is that each of the four long periods of meditation I experienced were times of great fulfilment, with nothing to suggest that something within me remained frustrated. However, if pots have exploded, it's because somewhere there was pressure, and that pressure must not have been there that day; it must have been there, somewhere out of my sight, for weeks or months, when I was intensely and totally absorbed in meditation.

But now I'm letting myself be carried away by the momentum of the pen (or rather, the typewriter). The reality is that (except in the last period of meditation, which was cut short by a combination of events and circumstances), the intensity of the meditation gradually diminished from one moment onwards, like a wave that was about to be followed by another that was about to take its place... The feeling of fulfilment, to tell the truth, followed this same movement, with the difference that it was only present at the time of the meditation-waves, and not the mathematical-waves.

The situation I am trying to define is no longer, it seems to me, a situation of conflict, but it is becoming apparent that it still contains the germ, the potentiality of conflict. For me, it is now perhaps the most visible sign of a division within me, through its impact on the course of my life. This division is none other than the boss-child division.

I can't put an end to it. All I can do, now that it has been clearly identified, is to pay attention to it, to follow its signs and its evolution over the months and years ahead. Perhaps this passion for maths, a little misguided it has to be said, will burn itself out (as another passion in me has already done...), to make way for the passion of discovering myself and my destiny.

This passion is vast enough, as I said, to fill my life - and surely my whole life will not be enough to exhaust it...

​ (50) The weight of a past

It's been a few days since I finished putting the finishing touches to Harvest and Sowing - after believing, for over a month, that I was [◊ 132] on the verge of finishing in the next few days. Even this time, after putting ‘the finishing touches’ on it, I wasn't entirely sure though if I had actually finished - there was still one question indeed that I had left hanging. It was ‘to understand what events or circumstances had finally triggered the “tipping” in the setting of the “boss”’, in favour of mathematics instead of meditation, against considerable forces of inertia. Without any deliberate intention, my thoughts returned with some insistence to this question, in these last few days when I had already begun to branch off into other matters of a completely different order, including mathematical questions (of conformal geometry). I might as well make the most of this ‘meditative end of the road’, to dig a little deeper and leave a clear space...

A number of associations arise when I try to answer ‘off the top of my head’ why ‘I'm getting back into maths’ (in the sense of a major investment that's likely to be long-term, of the order of at least a few years). Perhaps the strongest of all relates to the feeling of chronic frustration that I have come to experience in my teaching activity over the last six or seven years. There's this increasingly strong feeling of being ‘underemployed’, and even, very often, of investing myself and giving the best of myself for morose pupils who don't care what I have to give.

I see wonderful things to do everywhere, just waiting to be done. It's the things themselves that tell us what language to develop to understand them, and what tools to acquire to explore them. I can't help seeing them, simply because of regular contact with maths (at however modest a level) as a result of teaching, even at times in my life when my interest in maths is at its most marginal. Behind each thing you see, no matter how little you look, there are other beautiful things, which in turn cover and reveal others... Whether in maths or elsewhere, wherever you look with genuine interest, you see a richness revealed, a depth opened up that you guess is inexhaustible. The frustration I'm talking about is that of not being able to communicate this feeling of richness - of depth - to my students, even if it's only a spark of desire to explore at least what's right at their fingertips, to give it their all during the few months or years that they've decided to invest in a so-called ‘research’ activity anyway, for the purposes of [◊ 133] preparing for this or that degree. With the exception of two or three of the students I have had over the last ten years, it seems that the very idea of ‘giving it their all’ frightens them, that they prefer for months and years to stand idly by, stamping their feet, or painstakingly doing mole's work of which they know neither the ins and outs, as long as there is the diploma at the end. There's a lot to be said for this kind of paralysis of creativity, which has nothing to do with the existence or non-existence of ‘gifts’ or ‘faculties’ - and this goes back to the very beginning of my thinking, when I touched in passing on the underlying cause of such blockages. But that's not my point here, which is rather to note the state of chronic frustration that these situations, constantly repeated throughout these last seven years of teaching, have ended up creating in me.

The obvious way of ‘resolving’ such frustration, at least insofar as it is that of the ‘mathematician’ in me and not that of the teacher, is to do for myself at least some of those things that I despaired of seeing any of my students grasp at the end of the day. And that's what I've done here and there, whether it be occasional reflection lasting a few hours, or even a few days, on the fringes of my teaching activity, or during periods of intense mathematical craving (which sometimes came on like real explosions...), which could last for weeks or months. Occasional work of this kind, in fits and starts, could usually only give rise to a very first rough sketch of a question, and to a very fragmentary vision - it was rather a clearer vision of the work in perspective, whereas this work itself still remains to be done and, if it is better seen, only appears all the more burning. Two months ago I gave an overall sketch of the main themes that I had begun to take some measure of. It is the Outline of a Programme, to which I have already had occasion to allude, and which will finally be joined to the present reflection, to constitute together volume 1 of Mathematical Reflections.

It is quite clear that this ‘private’ exploration alone could not resolve my frustration. This feeling of ‘being underemployed’ surely reflected a desire (egotistical in origin, I believe, i.e. ‘the boss's’ desire) to take action. It's less a question here of acting on others (on my students, let's say, getting them going, ‘communicating something’ to them, or helping them to get such and such a diploma that might enable them to apply for such and such a job, etc.) than of acting ‘as a mathematician’: contributing to the discovery of such and such unsuspected facts, to the emergence of such and such a theory, and so on. This is immediately associated with the observation made earlier that mathematics is a ‘collective adventure’. If I ask myself how I felt when I was doing maths over the last ten years, at a time in my life when it would never have occurred to me that I might one day go back to publishing, and when it was also more or less clear that none of my present or future students would have anything to do with my prospecting work - it immediately became clear to me that these were by no means the dispositions of someone who would do something for personal pleasure alone, or driven by an inner need that concerned only himself, with no relationship to others.

When I do maths, I believe that somewhere inside me it is clearly understood that this maths is meant to be communicated to others, to be part of something wider to which I am contributing, something that is in no way individual in nature. I could call this ‘thing’ ‘mathematics’, or better still, ‘our knowledge of mathematical things’. The term ‘our’ here undoubtedly refers, first and foremost, in concrete terms, to the group of mathematicians whom I know and with whom I have interests in common; but there is also no doubt that it goes beyond this restricted group just as much as it goes beyond me personally. This ‘our’ refers to our species, insofar as it, through some of its members throughout the ages, has been and is interested in the realities of the world of mathematical objects. Before this very moment when I am writing these lines, I had never thought about the existence of this ‘thing’ in my life, and even less about questioning its nature and its role in my life as a mathematician and teacher.

The desire to take action to which I have alluded seems to me to take the following form in my life as a mathematician: to bring out of the shadows what is unknown to everyone, not just to me (as I saw earlier), and this, moreover, for the purpose of being made available to everyone, thus enriching a common ‘heritage’. In other words, it is the desire to contribute to the enlargement and enrichment of this ‘thing’, or ‘heritage’, which goes beyond my person.

In this desire, certainly, the desire to enlarge my person through my works is not absent. In this aspect, I find the craving for ‘growth’, [◊ 135] for enlargement, which is one of the characteristics of the self, of the ‘boss’; this is its invasive and, ultimately, destructive aspect (44’). Yet I also realise that the desire to increase the number of things that (for a short or long time) will more or less bear my name, is far from exhausting, from covering up this desire or this wider force, which drives me to want to contribute to enlarging a common heritage. It seems to me that such a desire could be satisfied (if not ‘in my company’, where the boss is still rather invasive, at least by a mathematician of greater maturity) while the role of my own person would remain anonymous. This would perhaps be a ‘sublimated’ form of the tendency to enlarge the ego through identification with something greater than oneself. Or perhaps this kind of force is not in itself egotistical, but more delicate and profound in nature, expressing a deep-seated need, independent of any conditioning, that attests to the profound link between the life of an individual and that of the whole species, a link that is part of the meaning of our individual existence. I don't know, and it's not my purpose here to explore such far-reaching questions.

Rather, my purpose is to examine (in a more modest way) a concrete situation concerning myself: a situation of frustration, with a partial and provisional outlet in the form of sporadic mathematical activity. The logic of the situation meant that sooner or later I would have to communicate what I found. Since, until last year, I was by no means prepared to make the large-scale, long-term investment in my mathematical passion that would have been necessary to ‘exploit’ the mines I was uncovering for publication purposes, by means of detailed ‘piecework’, I was left with the alternative of communicating to certain mathematician friends who were sufficiently ‘in the know’ at least those things that were closest to my heart.

I think that if, over the last ten years, I had found a mathematician friend who could act as an interlocutor and source of information for me (as Serre had done to a very large extent for many years in the 1950s and 1960s), as well as a relay for the ‘information’ that I could pass on to him (a role that Serre didn't have to play in the past, because I took care of it myself! ), my desire to ‘take action in maths’ would have found sufficient satisfaction to resolve my frustration, while contenting myself with an [◊ 136] episodic and moderate investment of energy in mathematics, leaving the lion's share to my new passion. The first time I approached a mathematician friend with such an expectation (at least implicit in me) was in 1975, and the last time in 1982, a year and a half ago. By an amusing coincidence, both times it was to try to ‘place’ (so that it could be passed on and, who knows, developed at the end of the ends! ) the same ‘programme’ of homological and homotopic algebra, the first seeds of which date back to the 1950s, and which was perfectly ‘mature’ (according to the intimate conviction I had of it) before the end of the 1960s; a programme whose preliminary development and broad outline is the very theme of this Pursuit of Fields, the Introduction to which I'm supposed to be writing at the moment! The fact remains that, for reasons that undoubtedly differ from one case to another, my attempts to rediscover a relationship of ‘privileged interlocutor’, such as there had been (before 1970) with Serre, and then with Deligne, came to nothing. One common factor, however, was the relatively limited time I was prepared to devote to maths. On the two occasions I've mentioned (in 1975 and 1982), this certainly contributed to making communication difficult. In fact, I was trying above all to ‘place’ something, without worrying too much about making the necessary effort to ‘bring myself up to speed’ so as to be a satisfactory interlocutor for my correspondent, who was much more ‘in the loop’ than I was (to say the least!) when it came to the techniques commonly used in homotopy.

I could consider the ‘Letter to...’ that serves as the first chapter of the Poursuite des champs (letter from February last year, just over a year ago) as my last attempt to find an echo, with one of my friends of yesteryear, for some of my ideas and concerns of today. The continuation of the reflection begun (or rather, taken up again) in that letter was to become (without my even suspecting it for weeks) the first mathematical text since 1970 to be promised publication. It was only almost a year later that I received an indirect reaction to this substantial letter (see note 38). It was more eloquent than any other letter received to date from a fellow mathematician, in making me feel certain attitudes towards my modest self that have become common among my mathematician friends since I left the milieu of which I was a part with them. There is in this letter, from one to whom I had addressed myself as a friend, in a disposition of warm sympathy, a deliberate purpose of [◊ 137] derision, which reminded me particularly violently of something of which I had come to realise myself more and more clearly during the last few years. Previously, I had had occasion above all to notice a distancing from myself, in the mathematical ‘great world’, and above all, among those who had been my more or less close friends (45). This is no longer a question of distancing oneself from people, but rather of a consensus, in the nature of a fashion and as it presents itself as something to be taken for granted, between people who are ‘in the know’ to some extent: that the kind of maths in thousand-page packets, and the notions with which I've been bending people's ears for a decade or two (46, 47), aren't very serious at all; that there's a lot of bombast there for not much worthwhile, and that apart from some general nonsense about the notion of schema and staggered cohomology (which do have their uses sometimes, alas, we're willing to admit), it's more charitable to at least forget the rest ; that those who would nevertheless pretend to still be singing this kind of Grothendieckian trumpet, despite good taste and the obvious canons of seriousness, are to be lumped in with their Master, avowed or not, and that they have only themselves to blame if they are treated as they deserve to be...

Surely the many echoes of this (which I have just transcribed ‘in plain English’) that have reached me since 1976 (50), and especially over the last two or three years, have finally awakened in me a fighting spirit that had become somewhat dormant over the last ten years. It's a completely idiotic reflex, like that of a bull to whom it's enough to show a piece of red cloth and wave it in front of his nose, for him to immediately get into a frenzy, forgetting the path he was following and which was his own! I still think that this reflex is quite epidermal, and that it wouldn't have been enough on its own to shake me up. Fortunately, doing maths has a lot more charm than running into a piece of cloth and getting larded from all sides. But doing maths, pursuing against all odds a style of work and an approach to things that are mine, also means ‘throwing myself into the fray’; it means asserting myself in the face of signs of disdain, of rejection - which come to me, no doubt, in response to the disdain that my former friends felt or thought they felt in me, if not towards them, at least towards a [◊ 138] milieu with which they continue to identify unreservedly. So it is also, if anything, following the piece of red cloth, instead of following my path.

This idea has occurred to me on several occasions over the last few weeks, and it is perhaps this aspect in particular that today's reflection has been directed towards. Along the way, another aspect has come to the fore, one in which the forces of the ego also play a large part, but which is not simply a combative reflex. Rather, a desire within me, the nature of which I cannot yet clearly discern, to give meaning to the mathematical work I have done over the last ten or twelve years, or to see it take on its full meaning; a meaning which (I am firmly convinced) cannot be reduced to that of private pleasure or personal adventure. But even if the nature of this desire remains unclear, since I haven't taken the time to examine it more closely, this reflection is enough to show me that it is really there, in this desire, that lies the force that weighs on me and forces my hand, so to speak, in favour of a mathematical investment - the force of ‘tipping’. It would work just as well, red fabric or not. If it's a sign of attachment to a past, it's the past of the last ten years, the ‘post-1970’ past, and not the past of things already written down in black and white, things done, things before 1970.

Basically, I'm not at all worried about these things, about the fate that the future, ‘posterity’, will have in store for them (although it's doubtful that there will even be a posterity...). What interests me in this past is not at all what I did (and the fortune that is or will be itss), but rather what was not done, in the vast programme that I had before my eyes at the time, and of which only a very small part was achieved by my efforts and those of the friends and students who sometimes kindly joined me. Without having planned or sought it, this programme itself has been renewed, along with my vision and my approach to mathematics. Over the years, the emphasis has shifted both in terms of themes and in terms of what I am trying to do: instead of accomplishing the great tasks of meticulous foundations, my primary aim now is to probe the mysteries that have fascinated me the most, such as that of ‘motives’, or that of the ‘geometric’ description of the Galois group of \(\bar{\Q}\)​ on \(ℚ\)​​. Along the way, of course, [◊ 139] I cannot help at least sketching out the foundations here and there, as I began to do (among other things) in ‘The Long Walk Through Galois Theory’, or as I am in the process of doing in the Pursuit of Fields. But the subject has changed, and so has the style that expresses it.

To put it another way: in the last ten years, I have glimpsed mysterious things of great beauty in the world of mathematics. These things are not personal to me, they are meant to be communicated - the very meaning of having glimpsed them, as I see it, is to communicate them, to be taken up, understood, assimilated... But communicating them, if only to oneself, also means going deeper into them, developing them a little - that's a job. I'm well aware, of course, that there's no question of my completing this job, even if I had a hundred years left to devote to it. But I don't have to worry about that now, about how many years or months I'm going to devote to this work out of the time I have left to live and discover the world, when there's another job waiting for me that only I can do. It's not in my power, and it's not my role, to regulate the seasons of my life.

NOTES FOR THE FIRST PART OF HARVEST AND SOWING

Contents

​ Note 1
​ Note 2
​ Note 3
​ Note 4
​ Note 5

My friends from Survivre et vivre

​ Note 6
​ Note 7
​ Note 8
​ Note 9
​ Note 10

Aldo Andreotti, Ionel Bucur

​ Note 11
​ Note 12
​ Note 13
​ Note 14
​ Note 15
​ Note 16
​ Note 17
​ Note 18

Jesus and the twelve apostles

​ Note 19
​ Note 20
​ Note 21
​ Note 22
​ Note 22

The child and the teacher

​ Note 23
​ Note 23'
Fear of playing

​ Note 23''

The two brothers

​ Note 23'''

Failure to teach (1)

​ Note \(23_{IV}\)

​ Note \(23_V\)
​ Note 24

Ethical consensus - and control of information

​ Note 25
​ Note 26

Youth snobbery’ - or the defenders of purity

​ Note 27
​ Note 28
​ Note 29
​ Note 30
​ Note 31

A hundred irons in the fire - or nothing to dry!

​ Note 32
​ Note 33

The powerless embrace
Note 34
Note 35
Note 36
Note 37
Note 38
Note 39

The visit

​ Note 40

Krishnamurti - or liberation turned hindrance

​ Note 41

The salutary uprooting

​ Note 42
​ Note 43
​ Note 44

NOTE 1 [◊ 141] (Added in March 1984) It is probably an overstatement to say that my ‘style’ and ‘method’ of working have not changed, while my style of expressing myself in mathematics has been profoundly transformed. Most of the time devoted over the last year to the Pursuit of Fields has been spent on my typewriter typing up thoughts that are destined to be published virtually as they are (apart from relatively short notes added later to make reading easier by cross-referencing, correcting errors, etc.). No scissors or glue to painstakingly prepare a ‘final’ manuscript (which, above all, must reveal nothing of the process that led to it) - that's a lot of changes in ‘style’ and ‘method’! Unless you dissociate the mathematical work itself from the work of writing and presenting the results, which is artificial, because it doesn't correspond to the reality of things, since mathematical work is indissolubly linked to writing.

NOTE 2 (Added in March 1984) When I reread these last two paragraphs, I had a certain feeling of unease, due to the fact that in writing them, I implicate others and not myself. Clearly, the thought that my own person might be involved hadn't occurred to me as I was writing. I certainly didn't learn anything when I confined myself to putting down in black and white (no doubt with a certain satisfaction) things that for years I have perceived in others, and seen confirmed in many ways. As I reflect further, I am led to remember that there has been no shortage of contemptuous attitudes towards others in my life. It would be strange if the link I have grasped between contempt for others and contempt for oneself were absent in the case of myself; sound reason (and also the experience of similar situations of blindness towards myself, which I have come to realise) tell me that this must surely not be the case! For the time being, however, this is no more than a simple deduction, the only possible use of which would be to encourage me to see with my own eyes what is going on, and to see and examine (if it does indeed exist, or has existed) this still hypothetical contempt for myself, so deeply buried that it has totally escaped my gaze until now. It's true that there has been no shortage of things to look at! [◊ 142] This one suddenly seems to me to be one of the most crucial, precisely because it is so hidden\(^{1}\)​...

NOTE 3 I am thinking here in particular of the famous conjectures of Mordell, Tate and Shafarevich, all three of which were demonstrated last year in a forty-page manuscript by Faltings, at a time when the well-established consensus of those ‘in the know’ ruled that these conjectures were ‘out of reach’! As it happens, the fundamental conjecture that serves as the cornerstone of the programme of ‘Anabelian algebraic geometry’ that is so dear to me is very close to Mordell's conjecture (it would even seem that the latter is a consequence of the former, which just goes to show that this programme is not a story for serious people).

NOTE 4 Even today, moreover, we come across ‘demonstrations’ of uncertain status. For years, this was the case with Grauert's proof of the finiteness theorem that bears his name, which no one (and there was no shortage of good people!) could read. This perplexity was resolved by other, more transparent demonstrations, some of which went further, taking over from the initial demonstration. A similar, more extreme situation is the ‘solution’ to the so-called ‘four-colour problem’, the computational part of which was solved using a computer (and a few million dollars). This is a ‘demonstration’ that is no longer based on an intimate conviction derived from an understanding of a mathematical situation, but on credit given to a machine that has no capacity for understanding, and whose structure and operation are unknown to the mathematician user. Even supposing that the calculation is confirmed by other computers, using other calculation programmes, I do not consider that the problem of the four colours is over. It will simply have changed its face, in the sense that it will no longer be a question of looking for a counter-example, but only a demonstration (readable, of course!).

NOTE 5 This fact is all the more remarkable because until about 1957 I was regarded with some reserve by more than one member of the Bourbaki group, which had ended up co-opting me, I believe, with some reluctance. A good-natured joke ranked me among the ‘dangerous specialists’ (in functional [◊ 143] analysis). I sometimes sensed in Cartan a more serious unspoken reserve - for some years I must have given him the impression of someone inclined towards gratuitous and superficial generalisation. I saw him quite surprised to find in the first (and only) rather long essay I wrote for Bourbaki (on the differential formalism of varieties) a reflection of any substance - he hadn't been too keen when I'd offered to take it on. (This reflection was useful to me again years later, when I developed the residue formalism from the point of view of coherent duality). I was more often than not left behind during the Bourbaki congresses, especially during the joint readings of the essays, being unable to keep up with the readings and discussions at the rate they were going. It's possible that I'm not really cut out for collective work. The fact remains that the difficulty I had in fitting into the group work, or the reservations I may have aroused for other reasons in Cartan and others, never attracted sarcasm or rebuke, or even a hint of condescension, except at most once or twice from Weil (definitely a case apart!). At no time did Cartan deviate from an equal kindness towards me, imbued with cordiality and also with that very special touch of humour that for me remains inseparable from his person.

​ My friends from Survivre et vivre

NOTE 6 Among these friends I should probably also count Pierre Samuel, whom I had known mainly in Bourbaki, like Chevalley, and who (like him) played an important role in the Survivre et vivre group. It doesn't seem to me that Samuel was much given to this illusion of the superiority of the scientist. Above all, I feel that he made a great contribution, through the common sense and smiling good humour he brought to joint work, discussions and relations with others, and also by gracefully taking on the role of the ‘ugly reformist’ in a group that was inclined towards radical analyses and options. He remained in Survivre et Vivre for some time after I withdrew, acting as editor of the newsletter of the same name, and left with good grace (to join Friends of the Earth) when he felt that his presence in that group had ceased to be useful.

​ Samuel belonged to the same restricted milieu as I did, but that didn't stop him from being one of the friends from those seething years from whom I think I learned something (bad pupil though I was...). His way of being [◊ 144], like Chevalley's even though they were hardly alike, was a better antidote for my ‘meritocratic’ tendencies than the most hard-hitting analysis!
It now seems to me that for all the friends from that period from whom I learnt something, it was more through their way of being and their sensitivity, which differed from mine, and from whom ‘something’ ended up being communicated, than through explanations, discussions, etc. I remember above all, in this respect, that I learnt a lot from them. In this respect, in addition to Chevalley and Samuel, I particularly remember Denis Guedj (who had a great influence on the Survivre et vivre group), Daniel Sibony (who kept his distance from this group, while pursuing its development with a half-disdainful, half-narcissistic eye), Gordon Edwards (who was a co-actor in the birth of the ‘movement’ in June 1970 in Montreal, and who for years did prodigious feats of energy to maintain an ‘American edition’ of the Survivre et vivre newsletter, in English), Jean Delord (a physicist about my age, a fine, warm-hearted man who took a liking to me and the Survrien microcosm), Fred Snell (another physicist based in the United States, from Buffalo, whose country house I stayed in for a few months in 1972).

​ Of all these friends, five are mathematicians, two are physicists, and all are scientists - which seems to show that the environment closest to me in those years remained an environment of scientists, and especially mathematicians.

NOTE 7 The preceding paragraph is the first in the entire introduction to be heavily crossed out in my original manuscript, and to be overwritten on numerous occasions. The description of the incident and the choice of words initially went against the grain, against the current - a force was clearly pushing to get over the incident quickly, as if by conscience, to ‘get down to business’. These are the familiar signs of resistance, here against the elucidation of this episode, and its significance as a revelation of an inner attitude. The situation is very similar to the one described at the beginning of this introduction (§ 2), that of the ‘crucial’ moment of the discovery of a contradiction and its meaning, in mathematical work: it is then the inertia of the mind, its reluctance to separate itself from an erroneous or insufficient vision (but one in which our person is in no way involved), which plays the role of [◊ 145] the ‘resistance’. The latter is of an active nature, inventive if necessary in order to succeed in drowning a fish even without water, whereas the inertia of which I have spoken is a merely passive force. In this case, even more than in the case of mathematical work, the discovery that has just appeared in all its simplicity, in all its obviousness, is followed in the moment by a feeling of relief from a weight, a feeling of liberation. It's not just a feeling - it's rather an acute and grateful perception of what has just happened, which is a liberation.

NOTE 8 As will become clear later, this ambiguity in no way ‘dissipated in the aftermath of the 1970 ‘revival’. This is a typical strategic retreat of the ‘I’, who writes off the period ‘before the awakening’, which immediately becomes the demarcation line for an irreproachable ‘after’!

NOTE 9 This is not entirely true; there is at least one exception among my closest colleagues, as will become clear later. There has been a typical ‘laziness’ of memory, which often tends to ‘overlook’ facts that do not ‘fit’ with a familiar and long-established view of things.

NOTE 10 For example, I have lost count of the number of letters, on mathematical as well as practical and personal matters, sent to colleagues or ex-students whom I considered to be friends, and which have never received a reply. It seems that this is not just special treatment for me, but a sign of a change in morals, according to echoes in the same vein. (Admittedly, these concern cases where the person sending a mathematical letter was not known to the recipient, who was a well-known mathematician...)

​ Aldo Andreotti, Ionel Bucur

NOTE 11 Of course, it's not impossible that I may have forgotten - not to mention that my particularly ‘polite’ disposition at the time would hardly encourage anyone to talk to me about such things, nor would it lead me to recall any such conversation that might well have taken place. What is certain is that it must have been very exceptional, to say the least, for the question of fear to be broached (without even calling it by that name...), and it must be just as exceptional today, especially in the ‘beau monde’.
[146] Of my many friends in that world, apart from Chevalley, who must have been aware of this atmosphere of fear at least during the 1960s, the only other person I can think of who must have perceived it clearly was Aldo Andreotti. I met him, his wife Barbara and their twin children (still very small) in 1955 (at a party at Weil's in Chicago, I think). We remained very close until the ‘great turning point’ in 1970, when I left the milieu that had been ours and lost sight of them for a while. Aldo had a very keen sensitivity, which had in no way been dulled by his dealings with mathematics and with the ‘polards’ like me. He had a gift for spontaneous sympathy for those he came into contact with. This set him apart from all the other friends I knew in the mathematical world, or even outside it. With him, friendship always took precedence over shared mathematical interests (of which there were plenty), and he is one of the rare mathematicians with whom I talked a little about my life, and he about his. His father, like mine, was Jewish, and he had suffered in Mussolini's Italy, as I had in Hitler's Germany. I saw him always available to encourage and support young researchers, in a climate where it was becoming difficult to be accepted by the establishment. His spontaneous interest was always in people, not in mathematical ‘potential’ or fame. He was one of the most engaging people I have ever had the good fortune to meet.

​ This mention of Aldo brings back memories of Ionel Bucur, who was also taken from us unexpectedly and before his time, and like Aldo, missed even more (I think) as a friend whom we loved to meet again, than as a partner in mathematical discussions. We sensed in him a kindness, alongside an uncommon modesty, a propensity to constantly take a back seat. It's a mystery how a man with so little inclination to think himself important or to impress anyone ended up as Dean of the Faculty of Sciences in Bucharest; no doubt because it never occurred to him to challenge the responsibilities that he was far from coveting, but which his colleagues or the political authorities were placing on his shoulders, which were, it has to be said, robust. He was the son of peasants (something that must have played a role in a country where ‘class’ is an important criterion), and had the common sense and simplicity of one. Surely he must have been aware of the fear that surrounds the man of notoriety, but surely it must also have seemed to him to go without saying, like the natural attribute of a position of power. I don't think, however, that he himself [◊ 147] ever inspired fear in anyone, certainly not in his wife Florica or their daughter Alexandra, nor in his colleagues or his students - and the echoes I have been able to get are very much in line with this.

NOTE 12 The word ‘tomorrow’ here should be taken literally, not as a metaphor.

NOTE 13 It is clear that the above description has no pretension other than to try to render as best I can, in concrete words, what this ‘fog’ of memory gives me, which has not condensed into any case of a kind that is even remotely precise, of which I could have given here a description that was even remotely ‘realistic’ or ‘objective’. It would be a misrepresentation to suggest from this passage that colleagues who are reluctant to sit in the front rows, or who do not have star or eminence status, are necessarily tied up in anguish when talking to one of them. This was visibly not the case for most of the friends I knew in this milieu, even among those who sometimes haunted conferences or seminars. What is unreservedly true is that the status of ‘eminence’ creates a barrier, a gulf vis-à-vis those without such status, and that it is rare for this gulf to disappear, even if only for the space of a discussion. I would add that the subjective distinction (which seems to me to be very real) between the ‘first ranks’ and the ‘marshes’ can in no way be reduced to sociological criteria (of social position, posts, titles, etc.) or even of ‘status’ or renown, but that it also reflects psychological particularities of temperament or dispositions that are more difficult to pin down. When I arrived in Paris at the age of twenty, I knew that I was a mathematician, that I had done maths, and despite the disorientation I've already mentioned, I felt that I was ‘one of them’, although I was the only one who knew it, and I wasn't even sure that I would continue to do maths. Today I'd be more inclined to sit in the back rows (on the rare occasions when the question arises).

NOTE 14 You might think that this contradicts the statement about the absence of a leader, but it does not. For the Bourbaki alumni, it seems to me that Weil was perceived as the soul of the group, but never as a ‘leader’. When he was there and when he liked it, he became the ‘ringleader’ as I said, but [148] he didn't lay down the law. When he was in a bad mood he could block discussion on such and such a subject that he disliked, even if it meant taking up the quiet subject again at another congress when Weil was not there, or even the next day when he was no longer obstructing. Decisions were taken unanimously by the members present, given that it was by no means out of the question (nor even rare) for one person to be in the right against the unanimity of all the others. This principle may seem absurd for group work. The extraordinary thing is that it actually worked!

NOTE 15 I did not get the impression that this ‘allergy’ to the Bourbaki style gave rise to communication difficulties between these mathematicians and myself or other Bourbaki members or sympathisers, as would have been the case if the spirit of the group had been that of a chapel, of an elite within the elite. Beyond styles and fashions, all the members of the group had a keen sense of mathematical substance, wherever it came from. It was only in the 1960s that I remember one of my friends referring to mathematicians whose work did not interest him as ‘troublemakers’. When it came to things about which I knew virtually nothing, I tended to take such assessments at face value, impressed by such casual confidence - until one day I discovered that this ‘pain in the arse’ was an original and profound mind, which did not please my brilliant friend. It seems to me that among certain Bourbaki members, an attitude of modesty (or at least reserve) towards the work of others, when one is unaware of that work or understands it imperfectly, eroded at first, while there still remained that ‘mathematical instinct’ which makes one feel a rich substance or a solid work, without having to refer to a reputation or a renown. From the echoes that reach me here and there, it seems to me that both modesty and instinct have become rare things today in what used to be my mathematical milieu.

NOTE 16 To tell the truth, several of the Bourbaki members surely had their own microcosm ‘of their own’, more or less extensive, apart from or beyond the Bourbakian microcosm. But it is perhaps no coincidence that, in my own case, such a microcosm did not form around me until after I had ceased to be part of Bourbaki, and all my energy had been invested in tasks that were personal to me.

NOTE 17 [◊ 149] It was above all outside the scientific community that I encountered warm echoes of the action to which I had committed myself, and active help. Apart from the friendly support of Alain Lascoux and Roger Godement, I must mention here above all that of Jean Dieudonné, who travelled to Montpellier for the court hearing, to add his warm testimony to others in favour of a lost cause.

NOTE 18 I believe that this lack of discernment did not stem from negligence on my part on those two occasions, but rather from a lack of maturity, an ignorance. It wasn't until about ten years later that I began to pay attention to blocking mechanisms, whether in my own person, in people close to me or in students, and to appreciate the immense role they play in everyone's life, and not just at school or university. Of course, I regret not having had the discernment of greater maturity on those two occasions, but not for having expressed my impressions clearly, whether well-founded or not. When, in a particular case, I found that work had been done without seriousness, it seemed to me to be a necessary and beneficial thing to name these things for what they are. If, in yet another case, the conclusion I drew was hasty and unfounded, I was not the only one whose responsibility was engaged. The pupil who had been shaken up in this way still had the choice of either learning from it (which is perhaps what happened on the first occasion), or allowing himself to be discouraged, and perhaps then changing profession (which is not necessarily a bad thing either!).

​ Jesus and the twelve apostles

NOTE 19 From 1970 to the present day, another student, Yves Ladegaillerie, has prepared and passed a thesis with me. The students of the first period were P. Berthelot, M. Demazure, J. Giraud. Mrs M. Hakim, Mrs Hoang Xuan Sinh. L. Illusie, P. Jouanolou. M. Raynaud, Mrs M. Raynaud, N. Saavedra, J.-L. Verdier. (Six of them completed their thesis work after 1970, at a time when my mathematical availability was extremely limited). Among these students, Michel Raynaud takes a special place, having found for himself the essential questions and notions which are the subject of his thesis work, which he moreover developed entirely independently; my role as ‘thesis director’ properly speaking was therefore limited to reading the finished thesis, constituting the jury and being a member of it.

​ [150] When it was I who proposed a subject, I was careful to limit myself to those to which I had a sufficiently strong relationship to feel able, if necessary, to support the student's work. One notable exception was Michèle Raynaud's work on local and global Lefschetz theorems for the fundamental group, formulated in terms of 1-fields on suitable scalar sites. This question seemed to me (and indeed turned out to be) difficult, and I had no idea of a proof for the conjectures I was proposing (which, incidentally, could hardly be doubted). This work continued in the early 1970s, and Mme Raynaud (as had previously been the case with her husband) developed a delicate and original method without any assistance from me or anyone else. This excellent work also opens up the question of extending Mme Raynaud's results to the case of n-fields, which seems to me to represent the natural outcome, in the context of schemes, of theorems of the ‘weak Lefschetz theorem’ type. The formulation of the conjecture relevant here (which can hardly be doubted either) nevertheless makes essential use of the notion of n-field, the pursuit of which is supposed to be the main object of the present work\(^{2}\) , as its name, À la poursuite des champs, indicates. We will no doubt come back to this in due course,

​ Another rather special case is that of Madame Sinh, whom I first met in Hanoi in December 1967, during a month-long seminar I gave at the evacuated Hanoi University. The following year I offered her the subject of her thesis. She worked in the particularly difficult conditions of wartime, her contact with me being limited to occasional correspondence. She was able to come to France in 1974/75 (on the occasion of the international congress of mathematicians in Vancouver), and to complete her thesis in Paris (before a jury chaired by Cartan, and including Schwartz, Deny, Zisman and myself).

​ Finally, I should mention Pierre Deligne and Carlos Contou-Carrère, both of whom were students of mine, the former around 1965-1968, the latter around 1974-1976. Both clearly had (and still have) unusual means, which they used in very different ways and with very different fortunes too. Before coming to Bures, Deligne had been a pupil of Tits (in Belgium) - I doubt that he had been a pupil of anyone in mathematics, in the usual sense of the term. Contou-Carrère had been a pupil of Santalo (in Argentina), and for a while of Thom (little [◊ 151] or so). Both of them already had the stature of a mathematician when the contact was established, except that Contou-Carrère lacked method and craft.

​ My mathematical role with Deligne was limited to informing him, on a piecemeal basis, of the little I knew about algebraic geometry, which he learnt as if he had always known it; and also, along the way, to raising questions to which he usually found answers, immediately or in the following days. These were the first works by Deligne that I knew. His work after 1970 (both for him and for my ‘official students’) is known to me only through very scattered and distant echoes\(^{3}\).

​ My role with Contou-Carrère, as he himself says at the beginning of his thesis, was limited to introducing him to the language of diagrams. In any case, I have only been remotely involved in the work he has been preparing as a doctoral thesis in recent years, on a highly topical subject which is beyond my remit.

​ It was after a few misadventures in the wide world that Contou-Carrère was finally led recently, in extremis and (it now seems to me) unwillingly, to call on my services to act as thesis director and to form a jury. (This exposed him to the risk of appearing as one of Grothendieck's students ‘after 1970’, in a conjecture where this can present serious disadvantages...). I carried out this task as best I could, and this will probably be the last time I carry out this function (at the level of a doctoral thesis). I am all the happier, in this rather special circumstance, for the friendly assistance of Jean Giraud, who also took a month or two of his time to do a thorough reading of the voluminous manuscript, on which he wrote a detailed and warm report.

NOTE 20 That reminds me of the subject Monique Hakim had taken up, which wasn't much more engaging, to tell the truth - I wonder how she managed to keep her spirits up! If she struggled at times, it was certainly not to the point of making her sad or sullen, and the work between us was done in a cordial and relaxed atmosphere.

NOTE 21 [◊ 152] It would perhaps be more accurate to say that for the temperament that is mine, it is the necessary maturity that I still lack to fully assume a teaching role. My acquired temperament has long been marked by an excessive predominance of ‘masculine’ (or ‘yang’) traits, and one aspect of maturity is precisely a ‘yin-yang’ balance with a ‘feminine’ (or ‘yin’) predominance.

​ (Added later.) Even more than maturity, I see that it's a certain generosity that I've lacked in my life as a teacher up to now - a generosity that expresses itself in a more delicate way than through availability of time and energy, and which is more essential. This lack did not manifest itself visibly (by an accumulation of situations of failure, let's say) in my first period of teaching, no doubt mainly because it was compensated for by a strong motivation in the students who chose to come and work with me. In the second period, on the other hand, from 1970 to the present day, it seems to me that this lack of motivation is at least one of the reasons, and in any case the one that involves me most directly, for the overall failure that I observe in my teaching at research level (from DEA level upwards). On this subject, see Outline of a programme, § 8, and § 9 ‘Assessment of a teaching activity’( Bilan d’une activité enseignante — translator's note), where the sense of frustration that this activity has left me with over the last seven or eight years shines through\(^{4}\).

NOTE 22 Perhaps not for much longer, since I have decided to apply for admission to the Centre national de la recherche scientifique, thus putting an end to my teaching activities in universities, which have become increasingly problematic over the last few years .

NOTE 22' Even after 1970, when my interest in maths became sporadic and marginal in my life, I don't think there was an occasion when I recused myself when a student called on me to work with him. I can even say that, apart from two or three cases, the interest of my post-1970 students in the work they were doing was far less than my own interest in their subject, even in the periods when I didn't care much about maths other than on the days when I went to university. So the kind of availability I had to [◊ 153] my pre-1970 pupils, and the extreme demand for work that was a main sign of it, would have made no sense to most of my later pupils, who did maths without conviction, as if by a continual effort they had to make on themselves....

​ The child and the teacher

NOTE 23 The term ‘transmit’ here does not really correspond to the reality of things, which reminds me of a more modest attitude. This rigour is not something that can be passed on, but at the very most it can be awakened or encouraged, whereas it is ignored or discouraged from an early age, by the family environment as well as by schools and universities. As far back as I can remember, this rigour has been present in my quests, those of an intellectual nature at least, and I don't think it was passed on to me by my parents, and even less by teachers, at school or among my mathematician elders. It seems to me to be one of the attributes of innocence, and therefore one of the things that everyone is born with. Very early on, this innocence ‘sees a lot of green and a lot of black’, which means that it is obliged to plunge more or less deeply, and that often there is hardly a trace of it in the rest of life. In my case, for reasons I haven't yet thought of investigating, a certain innocence has survived at the relatively benign level of intellectual curiosity, whereas everywhere else it has plunged deep, unseen and unheard! like everyone else. Perhaps the secret, or rather the mystery, of ‘teaching’ in the full sense of the term, is to rediscover this seemingly vanished innocence. But there is no question of rediscovering this contact in the pupil if it is not already present or rediscovered in the person of the teacher himself. And what is ‘transmitted’ by the teacher to the pupil is by no means this rigour or this innocence (innate in both of them), but a respect, a tacit revaluation of this commonly rejected thing.

NOTE 23' However, for the last seven or eight years there has been another chronic “source of frustration” in my life as a mathematician, but one that has expressed itself over the years in a much more discreet way. It ended up becoming apparent through an effect of repetition, of obstinate accumulation of the same type of ‘frustrating’ situation in my teaching activity, and finally bursting into a sort of ‘fed up!’, causing me to put an end to practically all so-called [◊ 154] ‘research direction’ activity. I touch on this question once or twice in the course of my reflection, and finally examine it at least a little at the very end. At the very least, I describe this frustration, and examine the role it played in my ‘return to maths’ (cf. § 50, ‘The weight of a past’).

​ Fear of playing

NOTE 23'' This student had worked with me on a DEA “work placement” for a whole year, and remained “contracted” in his working relationship with me right up to the end. It was a frankly friendly relationship, shot through with a mutual sympathy that could not be doubted. And yet there was this ‘stage fright’; this fear, the real cause of which was certainly not fear of me, although it looked like it. I might not even have noticed if this student hadn't told me about it himself, no doubt to ‘explain’ more or less the reason for an almost complete block in his work during the year.
As had happened with other students who, like him, had taken to a certain geometrical substance at the beginning, the blockage became apparent from the moment it was a question of doing ‘work on parts’, i.e. putting statements in black and white, or just grasping the meaning and significance of those that I provided and proposed to accept as the basis of a language, as the ‘rules of the game’. The ‘school’ reflex almost always pushes the student, faced with a situation in which he is supposed to be ‘doing research’, to adopt as a ‘given’, both vague and imperative, the implicit ‘rules of the game’ that are transmitted by the teacher, and which it is above all not a question of trying to explain, let alone understand. The concrete form these implicit rules take are ‘recipes’ for semantics or arithmetic, along the lines of, say, a mole book (or any other common textbook). (I don't think that the attitudes of most professional mathematicians, and of other scientists too, are essentially different - except that the ‘master’ is replaced by the ‘consensus’ that sets the rules of the game at the moment and considers it an immutable given. This consensus also lays down which ‘problems’ are to be solved, between which everyone feels free to choose as they wish, even allowing themselves to modify them in the course of their [◊ 155] work, or even to invent new ones...) I have noticed that my entirely different attitude towards a mathematical substance that has to be probed, and therefore also towards the pupil, almost certainly triggers disarray, one of the signs of which is anxiety. Like all anxiety, this will tend to take on a face, to project itself onto an external ‘reason’, plausible or not. One of the most common faces of anxiety is fear.

Such difficulties hardly arose in the first period of my teaching activity, except perhaps in the two cases where a ‘teacher-pupil’ relationship didn't continue beyond a few weeks, and perhaps (I can't say) in the case of the ‘sad pupil’, who perhaps felt ‘riveted’ to a subject that didn't inspire him at all, even though he had every opportunity to change it. In the case of the student (whom I also mentioned) who remained afflicted by stage fright for a long time, it's clear that the reason lay elsewhere. He was in no way blocked in his work, but on the contrary was perfectly at ease with the theme he had chosen, on which he had done a great deal of groundwork. Most of my students during this period were also former students of the École normale, and their contact with Henri Cartan had already shown them the example of a ‘different’ approach to mathematics. At the opposite end of the spectrum (so to speak), in my second period as a teacher, at the University of Montpellier, it was among the first-year students that the anxiety I have mentioned least interfered with the work of reflection. For many of these students, the astonishment of a different approach did not provoke anguish or closure, but on the contrary openness and enthusiasm to do interesting things for once! From my observations, the effect of a few years at university on a student's creative disposition is radical and devastating. It's a strange thing that in this respect the effect of the long years of high school seems relatively innocuous. Perhaps the reason is that the university years come at an age when the creativity innate in us must ultimately be expressed through personal work, otherwise we will be shipwrecked forever, at least as far as creative work of an intellectual nature is concerned. It was surely by a healthy instinct that during my student years (also at Montpellier University) I practically refrained from setting foot in lectures, devoting almost all my energy to personal mathematical reflection [◊ 156].

​ The two brothers

NOTE 23’‘’ This student's antagonism took the form, from the outset, of a ‘class antagonism’: I was the ‘boss’ who had ‘power of life and death’ over his mathematical future, which I could decide at my whim... Of course, events could only confirm this vision, since I was quick to put an end to my responsibilities (which had become painful) towards this student. This put him in a tricky situation, in these times when it's not so easy to find a ‘boss’, especially when the subject has already been chosen. For the other student, frustrated in his legitimate expectations, the antagonism took a similar form. I was seen as the tyrannical ‘mandarin’ who could not tolerate any contradiction from those (students or lower-ranking colleagues) whom he considered to be his subordinates.

Such a ‘class attitude’ never manifested itself, if at all, during the relationship with my students in the first period. The obvious reason was that in the pre-1970 context, there was no doubt that once the student had passed his thesis, he would have a post as a lecturer, and would therefore enjoy a social status identical to mine, that of ‘university professor’. The figures are revealing: the eleven students who began working with me before 1970 were appointed lecturers as soon as they had completed their work, whereas none of the twenty or so students who worked more or less under my supervision were appointed to such a post. It is true that only two of them were motivated enough to do a doctoral thesis (an excellent one in both cases).

So it's not surprising that in this second period, certain ambivalences (whose deep origins remained hidden) took the form of class antagonism and distrust (presented and felt as ‘visceral’) towards the ‘boss’. For one of those who had been more or less a pupil, friendly relations continued for about ten years, without any apparent antagonistic episode, and yet marked by this same ambiguity, expressed in an attitude of mistrust, held ‘in reserve’ behind manifest sympathy. To tell the truth, I was never fooled by this commanding ‘mistrust’, which appeared to me above all as a reason that this friend thought he should give himself for not venturing outside the well-defined domain that he [◊ 157] had chosen as his own, in his professional life as in his life quite simply - something that he was free to do, however, without anyone (except, at most, himself!) calling him to account....

These three cases are the only ones, in all my teaching experience, where a certain ambivalence in the relationship between a pupil (or someone who is more or less a pupil) and myself has been expressed by a ‘classroom attitude’. Such an attitude appears particularly ambiguous when it manifests itself between colleagues within an academic ‘body’ where they both enjoy exorbitant privileges compared to the situation of ordinary mortals, privileges which make differences in rank (and salary) appear relatively insignificant. I have noticed that these attitudes disappear as if by magic (and with good reason!), as soon as the person concerned sees himself promoted to the position of which only the day before he was complaining to others.

I detect a similar ambiguity in most, if not all, of the conflict situations I have witnessed within the mathematical world (and often outside it too). Those who are ‘cased’, whether or not their rank corresponds to their expectations (justified or not), enjoy quite unheard-of privileges that no other profession or career can offer. Those who don't have a job aspire to the same security and the same privileges (which doesn't necessarily prevent them from taking an interest in maths itself, and sometimes from doing great things). These days, when the competition to fit in is fierce and the unmatched are often treated like laggards, I've more than once felt the connivance between the person who enjoys humiliating and the person who is humiliated - and who swallows and crushes. The real object of his bitterness and animosity is not the one who has used power, but none other than himself, who has crushed himself and invested the other with this power that he uses to his pleasure. The one who takes pleasure in humiliating is also the one who takes revenge and compensates (without ever erasing it...) for a long-lasting humiliation that has long been buried and forgotten. And he who acquiesces in his own humiliation is his brother and emulator, who secretly envies it and in bitterness buries both the humiliation and the humble message about himself that it brings him.

​ Failure to teach (1)

NOTE \(23_{IV}\) Since these lines were written, I have had occasion to speak [◊ 158] with two of my ex-students from after 1970, in an attempt to probe with them the reason for the failure of my teaching at research level at the University of Montpellier. They told me that my tendency to underestimate the difficulty that the assimilation of such techniques, familiar to me but not to them, could represent for them, had had a discouraging effect on them, because they constantly felt that they were falling short of the expectations I had of them. What's more (and this seems to me to be of even greater significance), they sometimes felt frustrated when I ‘sold them out’ by giving them a shaped statement that I had up my sleeves, instead of letting them have the pleasure of discovering it on their own, at a time when they were already very close to it. After that, all they had to do was the ‘exercise’ (which they weren't otherwise keen on) of proving the statement in question. This is where the ‘lack of generosity’ in me that I had noted in an earlier note (note 21) comes in, without going into further detail on the subject. It is disappointments like these, above all, that represent my personal contribution to the disappearance of interest in research in both of them, after what was nonetheless an excellent start.

I realise that I was no more generous before 1970 than I was after. If I didn't have the same difficulties then, it's probably because the kind of students who came to me at that time were motivated enough to find even a ‘long exercise’ appealing, which was an opportunity to learn the trade and a whole host of other things along the way; and also, for a starter statement that I was ‘selling the fuse’ on, to come up with a whole host of others on their own that went far beyond the first. When I changed teaching location, I made the necessary adjustment in the choice of topics for reflection that I proposed to my new students, by choosing mathematical objects that could be grasped by immediate intuition, independently of any technical baggage. But this essential adjustment was in itself insufficient, because of differences in attitude (in my new pupils compared with those of yesteryear), which were even more important than a single difference in background. This also ties in with the observation made earlier (beginning of § 25) about a certain inadequacy in me for the role of ‘master’, which came out much more strongly in my second period as a teacher than in the first.

NOTE \(23_V\) [◊ 159] A particularly striking sign of this difference came on the occasion of the ‘stranger episode’, of which I have had occasion to speak (Section 24). While I received expressions of sympathy from many people who were complete strangers to me, I don't remember any of my pre-1970 students thinking of expressing themselves in this way, let alone offering me any help in the action I had embarked upon. On the other hand, it seems to me that there is not one of my students or former students from the second period who did not express their sympathy and solidarity with me, and several of them took an active part in the campaign I was conducting at local level. Beyond this restricted circle, the 1945 ordinance affair also created a certain amount of emotion among many students at the faculty who knew me by name at most, and a good number of them came to the courthouse on the day I was summoned to show their solidarity. This last circumstance suggests, moreover, that the difference I observed between the attitudes of my students ‘before’ and ‘after’ 1970 is perhaps less an expression of the difference in relations between them and me than a difference in mentalities. Clearly, my ‘before’ students had become important people, and it takes a lot for important people to consent to be moved... But the episode of my departure from IHES in 1970 and my involvement in militant action seems to show that it's not just that. It was a time when none of them was yet such an important figure, and yet I don't remember any of them showing the slightest interest in the activity I was getting involved in. I rather think that it must have made them uncomfortable, all of them without exception. This again points to a difference in mentality, but one that cannot be blamed solely on differences in social status.

NOTE 24 The ethics I am talking about apply just as much to any other environment formed around research activity, where the possibility of making one's results known, and of taking credit for them, is a matter of ‘life and death’ for the social status of any member, or even of ‘survival’ as a member of that environment, with all the consequences that implies for him and his family.

​ Ethical consensus - and control of information

NOTE 25 Apart from the conversation with Dieudonné, I cannot recall a conversation in which I was a participant or witness, during my life as a [◊ 160] mathematician, in which the ethics of the profession were discussed, the ‘rules of the game’ in relations between members of the profession. (I exclude here the discussions about the collaboration of scientists with military apparatuses, which took place in the early 1970s around the Survive and Live movement. They didn't really concern the relationships between mathematicians. Many of my friends in Survivre et vivre, including Chevalley and Guedj, felt that the emphasis I placed at that time, especially in the early days, on this question to which I was particularly sensitive, distracted me from more essential everyday realities, of precisely the kind I am examining in this reflection). These things were never discussed between a student and myself. The tacit consensus was limited, I think, to this one rule, not to present as one's own the ideas of others of which one may have become aware. This is a consensus, it seems to me, that has existed since Antiquity and has not been challenged in any scientific milieu to this day. But in the absence of this other complementary rule, which guarantees all researchers the possibility of making their ideas and results known, the first rule remains a dead letter. In today's scientific world, men in positions of prestige and power have discretionary control over scientific information. In the environment I knew, this control is no longer tempered by a consensus like the one Dieudonné was talking about, which perhaps never existed outside the restricted group whose spokesman he was. The scientist in a position of power receives practically all the information he deems useful to receive (and often even more), and he has the power, for a large part of this information, to prevent its publication while retaining the benefit of the information received and rejected as ‘uninteresting’, ‘more or less well known’, ‘trivial’, and so on. I return to this situation in note 27.

NOTE 26 The ‘founding members’ of Bourbaki were Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné and André Weil. They are all still alive, with the exception of Delsarte, who died before his time in the 1950s, at a time when the ethics of the profession were still generally respected.

When I reread the text, I was tempted to delete this passage, in which I may give the impression of awarding certificates of ‘probity’ (or non-probity) that the people concerned have no use for, and that it is not my job [◊ 161] to do. The reservation that this passage may arouse is surely justified. I have nevertheless kept it, for the sake of the authenticity of the testimony, and because this passage does indeed convey my feelings, even if they are misplaced.

​ Youth snobbery’ - or the defenders of purity

NOTE 27 Ronnie Brown shared with me a comment by J. H. C. Whitehead (of whom he was a pupil) about the ‘snobbery of the young, who believe that a theorem is trivial because its proof is trivial’. Many of my old friends would do well to reflect on these words. This ‘snobbery’ is by no means limited to young people today, and I know more than one prestigious mathematician who practises it routinely. I'm particularly sensitive to it, because the best work I've done in mathematics (and elsewhere too...), the notions and structures I've introduced that seem to me to be the most fruitful, and the essential properties I've been able to derive from them through patient and persistent work, all fall under this label of ‘trivial’. (None of these things would nowadays have much chance of being accepted for a grade in the CRs, if the author were not already a celebrity!) My ambition as a mathematician throughout my life, or rather my passion and my joy, have constantly been to find the obvious things, and this is my sole ambition also in the present work (including in this introductory chapter...). The decisive thing is often already to see the question that had not been seen (whatever the answer may be, and whether it has already been found or not) or to come up with a statement (even if it is conjectural) that sums up and contains a situation that had not been seen or understood; if it is demonstrated, it matters little whether the demonstration is trivial or not, which is entirely incidental, or even whether a hasty and provisional demonstration proves to be false. The snobbery of which Whitehead speaks is that of the jaded wine-lover who does not deign to enjoy a wine until he is sure that it has cost a lot of money. More than once in recent years, caught up in my old passion, I have offered the best I had, only to see it rejected by this smugness. I have felt a pain that remains alive, a joy that has been disappointed - but that doesn't mean I'm homeless, and fortunately for me I wasn't trying to fit in an article of my own.
The snobbery of which Whitehead speaks is an abuse of power and a dishonesty, not only an insensitivity or a closure to the beauty of things, when it is exercised by a man of power against a researcher at his [◊ 162] mercy, whose ideas he has free rein to assimilate and use, while blocking their publication on the pretext that they are ‘obvious’ or ‘trivial’, and therefore ‘of no interest’. I am not even thinking here of the extreme situation of plagiarism in the ordinary sense of the term, which must still be very rare in mathematical circles. However, from a practical point of view, the situation is the same for the researcher who pays the price, and the inner attitude that makes it possible does not seem to me to be very different either. It is simply more comfortable, while it is accompanied by a feeling of infinite superiority over others, and by the good conscience and intimate satisfaction of the person who poses as the intransigent defender of the intangible purity of mathematics.

NOTE 28 In writing the preceding pages, I was initially divided between the desire to ‘get it off my chest’ and a concern for reserve or discretion. I had therefore remained somewhat vague, which was surely the main reason for my unease, for my feeling that ‘I wasn't learning anything’. Since the lines noting this malaise were written, I have twice rewritten those pages that had left me feeling so discontented, by getting more involved and getting to the bottom of things. In the process, I did indeed end up ‘learning something’, and I also believe that at the same time I managed to put my finger on something important, which goes beyond both the case in point and my own person.

NOTE 29 By this I mean an intense, long-term investment in mathematics, or in some other entirely intellectual activity. On the other hand, the deployment of such a passion, which can be a way of reacquainting ourselves with a forgotten force within us, and an opportunity to measure ourselves against a reluctant substance and, in the process, to renew and enrich our sense of identity with something that is truly personal to us - such a deployment can very well be an important stage in an inner journey, in a maturing.

NOTE 30 For some years now, it has been my children who have taken over, teaching a sometimes reluctant pupil about the mysteries of human existence....

NOTE 31 I'm thinking here of the ‘yang’ form of the desire to know - the one who probes, [◊ 163] discovers, names what appears... It is having been named that makes the knowledge that has appeared irreversible, indelible (even if it is later buried, forgotten, ceases to be active...). The ‘yin’, ‘feminine’ form of the desire for knowledge lies in an openness, a receptivity, in a silent welcoming of a knowledge that appears in the deepest layers of our being, where thought has no access. The appearance of such openness, and of a sudden knowledge that for a time erases all trace of conflict, comes as a grace once again, that touches deeply even though its visible effect may be ephemeral. I suspect, however, that this wordless knowledge that comes to us in this way, at certain rare moments in our lives, is just as indelible, and its action continues even beyond the memory we may have of it.

​ A hundred irons in the fire - or nothing to dry!

NOTE 32 When I was still doing functional analysis, i.e. up until 1954, I used to persist endlessly on a question that I couldn't solve, even though I had no more ideas and was content to go round in circles with old ideas that obviously didn't ‘bite’ any more. In any case, this was the case for a whole year, particularly for the ‘approximation problem’ in topological vector spaces, which was only to be solved some twenty years later by methods of a completely different order, which could only have escaped me at that point. I was driven then, not by desire, but by stubbornness, and by an ignorance of what was going on inside me. It was a painful year - the only time in my life when doing maths had become painful for me! It took that experience for me to realise that there's no point in ‘skipping’ - that once a piece of work has come to a standstill, and as soon as you realise you've come to a standstill, you have to move on to something else - even if it means coming back to the question at hand at a more propitious moment. This moment almost always comes quickly - the question matures, without me even pretending to touch it, simply by virtue of working enthusiastically on questions that may seem to have nothing to do with it. I'm convinced that if I persisted, I wouldn't get anywhere even in ten years! It was from 1954 onwards that I got into the habit in maths of always having many irons in the fire at the same time. I only work on one of them at a time, but by a kind of miracle that is constantly renewed, the work I do on one [◊ 164] also benefits all the others, which are biding their time. It has been the same, without any deliberate intention on my part, from my first contact with meditation - the number of burning questions to be examined has increased day by day, as reflection has continued....

NOTE 33 This is not to say that the moments in the work when paper (or the blackboard, which is a variant of it) is absent are not important in mathematical work. This is especially true in the ‘sensitive moments’ when a new intuition has just appeared, when it is a question of ‘getting to know’ it in a more global, more intuitive way than by ‘working on the pieces’, which this informal stage of reflection prepares. In my case, this kind of reflection is mostly done in bed or on walks, and it seems to me that it represents a relatively modest proportion of the total time devoted to the work. The same observations apply to meditation work as I've practised it so far.

​ The powerless embrace

NOTE 34 The word ‘embrace’ is by no means a mere metaphor for me, and the common language here reflects a profound identity. It might be said, not without reason, that it is not true then that embrace without wonder is powerless - that the earth would be depopulated if not deserted, if it were so in the literal sense. The extreme case is that of rape, in which wonder is certainly absent, even though a being may be procreated in the raped woman. Of course, the child born of such an embrace cannot fail to bear the mark of it, which will be part of the ‘package’ that it receives as a share and which it is up to it to assume; this does not prevent a new being from being conceived and born, from being created, a sign of power. And it's also true that sometimes a mathematician, whom I've seen full of self-importance, finds and proves beautiful theorems, signs of an embrace that was not lacking in strength! But it is also true that if the life of such a mathematician is suffocated by his smugness (as was to some extent the case in my own life, at one time), the fruits of these embraces with mathematics are of no benefit to him or to anyone else. And the same can be said of the father and mother of a child born of rape. If I speak of an ‘embrace without strength’, I mean above all the powerlessness to engender renewal in the person who believes he is creating, when in fact he is only creating a product, something outside himself, with no deep resonance within himself; a [◊ 165] product which, far from liberating him, creating harmony within him, binds him more closely to the fatuity within him of which he is a prisoner, which ceaselessly pushes him to produce and reproduce. This is a form of powerlessness at a deep level, behind the appearance of ‘creativity’ which is basically just unbridled productivity.

​ I've also had ample opportunity to realise that complacency, the inability to marvel, is in the nature of a real blindness, a blockage of a natural sensitivity and flair; if not a total and permanent blockage, at least one that is manifest in certain situations. It is a state in which a prestigious mathematician sometimes reveals himself, in the very things in which he excels, to be as stupid as the most stubborn of schoolchildren! On other occasions he will perform prodigious feats of technical virtuosity. I doubt, however, that he is yet in a position to discover the simple and obvious things that have the power to renew a discipline or a science. They are far too far below him for him to deign to see them! To see what no one else deigns to see requires an innocence that he has lost, or banished... It is surely no coincidence, given the prodigious increase in mathematical production over the last twenty years, and the bewildering profusion of new results that overwhelm mathematicians who simply want to ‘keep up to date’, that there has hardly been (as far as I can judge from the echoes that reach me here and there) any real renewal, any large-scale transformation (and not just by accumulation) of any of the major themes of thought with which I have been at all familiar.

​ Renewal is not a quantitative thing, it is foreign to a quantity of investment, measurable in a number of mathematician-days devoted to a given subject by such and such mathematicians of such and such a ‘level’. A million mathematician-days is powerless to give birth to something as childlike as the zero, which has renewed our perception of number. Only innocence has this power, a visible sign of which is wonder...

NOTE 35 This ‘gift’ is no one's privilege; we are all born with it. When it seems to be absent in me, it's because I've chased it away myself, and it's up to me to welcome it back. In me or in such-and-such, this ‘gift’ expresses itself in a different way than in such-and-such, in a less communicative way, less irresistible perhaps, but it is no less present, and I wouldn't know whether it is less active.

NOTE 36 [◊ 166] Such a delicate sensitivity to beauty seems to me intimately linked to something I have had occasion to speak of as ‘exigency’ (with respect to oneself) or ‘rigour’ (in the full sense of the term), which I described as ‘attention to something delicate in ourselves’, attention to a quality of understanding of the thing being probed. This quality of understanding of a mathematical thing cannot be separated from a more or less intimate, more or less perfect perception of the ‘beauty’ particular to that thing.

NOTE 37 I hardly need add, I think, that this long-term work has brought to light, day by day, much more than the ‘result’ I have just delivered in lapidary form. It's no different for a work of meditation than for a mathematical work motivated by a particular question that we set out to examine. Very often the twists and turns of the road followed (which may or may not lead to a more or less complete clarification of the initial question) are more interesting than the initial question or the ‘final result’.

NOTE 38 These notes were in fact a continuation of the long letter to..., which became the first chapter. They were typed so as to be legible for this old friend, and for two or three others (Ronnie Brown in particular) whom I thought might be interested. This letter, incidentally, was never answered, nor was it read by the addressee, who almost a year later (when I asked him if he had received it) expressed sincere astonishment that I should have thought for a moment that he could read it, given the kind of mathematics that was to be expected of me...

NOTE 39 This was the period, among other things, of the ‘long march through Galois theory’, referred to in Esquisse d'un programme (§ 3: ‘ fields of numbers associated with a child's drawing’).

​ The visit

NOTE 40 The work on this dream is the subject of a long letter in English to a friend and colleague who had dropped by my house the day before. Some of the materials used by the Dreamer, to bring this strikingly realistic dream out of apparent nothingness, were obviously borrowed from this short episode of the visit of a dear friend whom I had not seen for nearly ten years. [167] So, on the first day of work and against my previous experience, I thought I could conclude that the dream that had come to me concerned my friend more than it concerned me - that it was he who should have had the dream and not me! It was a way of evading the message of the dream, which (I should have known from the start from my past experience) concerned no one but me. I finally realised this during the night that followed this first, superficial phase of the work, which I resumed the next day in the same letter. Since that memorable letter, I have received no sign of life from this friend, one of the closest I have ever had.

​ This work was the only meditation that took the form of a letter (and in English to boot), so I no longer have any written trace of it. I was particularly struck by this episode, one of many that show the extent to which any sign of work that goes beyond a certain façade, and brings to light simple facts that we generally make a point of ignoring - the extent to which any such work inspires unease and fear in others. I will come back to this later (see § 47, ‘The solitary adventure’).、

​ Krishnamurti - or liberation turned hindrance

NOTE 41 It would be inaccurate to say that the only thing I took away from this reading was a certain vocabulary, and a propensity to make it my own and ultimately substitute it, appropriately enough, for reality. The reason I was so struck by Krishnamurti's first book (and even then I only had time to read a few chapters) was that what he was saying totally overturned a number of things that I took for granted, and which I immediately realised were commonplaces that had always been part of the air I'd breathed. At the same time, this reading drew my attention, for the first time, to far-reaching facts, especially that of flight from reality, as one of the most powerful and universal conditioning of the mind. This gave me an essential key to understanding situations that until then had been incomprehensible and therefore (without my realising it until I discovered meditation five or six years later) a source of anxiety. I was immediately able to see the reality of this flight all around me. This relieved certain anxieties, without however changing anything essential, because I only saw this reality in others, while at the same time [◊ 168] telling myself (as a matter of course) that it didn't exist in myself, that I was in short the exception that confirmed the rule (and without asking myself any other questions about this truly remarkable exception). In fact, I was in no way curious about anyone else or myself. This ‘key’ can only open in the hands of someone with the desire to penetrate. In my hands it had become an exorcism and a pose.

​ It was at the beginning of 1974 that for the first time I realised that the destruction in my life, which was following me step by step, could not have come from others alone, that there was something in me that attracted it, fed it and perpetuated it. It was a moment of humility and openness, conducive to renewal. But it remained peripheral and ephemeral, because I didn't work on it in depth. That ‘something inside me’ was still vague. I could see that it was a lack of love, but the very idea of working to identify more closely where and how there had been a lack of love in me, how it had manifested itself, what its concrete effects had been, etc., such an idea couldn't come to me, nor could it be put into practice. - (On the contrary, K. likes to insist on the vanity of all work, which he automatically equates with the ego's ‘craving to become’). So, with a borrowed ‘wisdom’ as my compass, I saw nothing else to do but wait patiently for ‘love’ to descend upon me like a grace from the Holy Spirit.

​ However, the humble truth that I had just learnt in the depths of a wave had given rise to a powerful wave of new energy, comparable to the one that would carry me two and a half years later into my first foray into meditation. This energy did not remain entirely unused. A few months later, when I was immobilised by a providential accident, it led to a (written) reflection in which, for the first time in my life, I examined the vision of the world that had been the unspoken basis of my relationship with others, and which came to me from my parents and especially my mother. It was then that I realised very clearly that this vision had failed, that it was incapable of accounting for the reality of relationships between people, and of fostering personal fulfilment in myself and in my relationships with others. This reflection remains marked by the ‘Krishnamurti style’, and also by the Krishnamurtian taboo on any real work towards understanding. [169] It did, however, make tangible and irreversible a knowledge that had been born a few months earlier, and which had initially remained vague and elusive. This knowledge, no book nor any other person in the world could then have brought it to me.

​ To have the quality of meditation, this reflection lacked above all a look at my own person and my vision of myself, and not just my vision of the world, a system of axioms in which I was not really ‘flesh and blood’. It also lacked a look at myself in the moment, at the very moment of reflection (which fell short of a real work); a look that would have allowed me to detect both a borrowed style and a certain complacency in the literary aspect of these notes, a lack of spontaneity and authenticity. As inadequate as it was, and relatively limited in its immediate effects on my relationships with others, this reflection nevertheless seemed to me to be a step, probably necessary given the starting point, towards the more profound renewal that was to take place two years later. It was then, at last, that I discovered meditation - by discovering that first unsuspected fact: that there were things to discover about myself - things that almost completely determined the course of my life and the nature of my relationships with others...

​ The salutary uprooting

NOTE 42 The ‘percussive’ event in question was the discovery, at the end of 1969, that the institution of which I felt a part was partly financed by funds from the Ministry of the Armed Forces, something which was incompatible with my basic axioms (and still is, in fact). This event was the first in a whole chain of others (each more revealing than the last!) which resulted in my leaving the IHES (Institut des hautes études scientifiques), and one thing leading to another in a radical change of environment and investments.

​ During the heroic years of the IHES, Dieudonné and I were the only members, and also the only ones to give it credibility and an audience in the scientific world, Dieudonné through the publication of Publications mathématiques (the first volume of which appeared in 1959, the year after Léon Motchane founded the IHES), and I through the séminaires de géométrie algébrique. In those early years, the existence of the IHES was very [◊ 170] precarious, with uncertain funding (from the generosity of a few companies acting as patrons) and with the only premises a room lent (with visible bad humour) by the Fondation Thiers in Paris for the days of my seminar$ ^5$ . I felt a bit like a ‘scientific’ co-founder, with Dieudonné, of my home institution, and I intended to live out my days there! I had come to identify strongly with the IHES, and my departure (as a consequence of the indifference of my colleagues) was experienced as a kind of uprooting from another ‘home’, before proving to be a liberation.

​ With hindsight, I realise that there must already have been a need for renewal in me, although I can't say how long ago it was. It's surely no mere coincidence that the year before I left the IHES, there was a sudden shift in my investment of energy, leaving behind the tasks that were still burning in my hands the day before, and the questions that fascinated me the most, to launch myself (under the influence of a biologist friend, Mircea Dumitrescu) into biology. I was embarking on this with a view to making a long-term investment in the IHES (which was in keeping with the multidisciplinary vocation of this institution). Surely this was merely an outlet for the need for a much more profound renewal, which could not have been achieved in the ‘scientific incubator’ atmosphere of the IHES, and which took place during the ‘cascade of awakenings’ to which I have already alluded. There have been seven, the last of which took place in 1982. The ‘military funds’ episode was providential in triggering the first of these ‘awakenings’. The Ministry of the Armed Forces, like my former colleagues at IHES, finally had my full gratitude!

NOTE 43 ‘The poetic work of my own composition’ contains many things that I know first-hand, and which today seem to me to be just as important in my life, and ‘in life’ in general, as when it was written, with the intention of publishing it. If I refrained from doing so, it was mainly because I realised later that the form was afflicted by a deliberate intention to ‘make poetic’, so that its conception of [◊ 171] an overly constructed whole, and many passages, lack spontaneity, to the point at times of a painful stiffness or swelling. This form, bombastic at times, was a reflection of my disposition, where decidedly it was often the ‘boss’ who led the dance - heavily, it goes without saying....

NOTE 44 It goes without saying that I am disregarding here the hypothesis, by no means improbable to say the least, of the unexpected eruption of an atomic war or some other similar event, likely to put an abrupt end once and for all to the collective game called ‘Mathematics’, and to many other things along with it...

  1. (August 1984) On this subject, however, see the reflection in the last two paragraphs of the note ‘The Massacre’, No. 87.
  2. This is in fact volume 3 of Mathematical Reflections, and not the present volume 1, Harvest and Sowing — see Introduction, p. V.
  3. In particular, I had the opportunity to go through some separate editions of Berthelot and Deligne, which they were kind enough to send me.
  4. Compare also note \(23_{IV}\) , added later.
  5. A recent pamphlet published by IHES on the occasion of the twenty-five anniversary of its foundation (of which Nico Kuiper was kind enough to send me a copy) does not breathe a word about these difficult beginnings, judged perhaps unworthy of the solemnity of The occasion, celebrated with great pomp last year.
posted @ 2025-03-12 18:21  TianyiLemon  阅读(40)  评论(0)    收藏  举报