5. Work and discovery

Contents

5.1.(1)The child and the good god…………………………..127

5.2.(2)Error and discovery………………………………..128

5.3.(3)The unavowable labors……………………………….129

5.4.(4)Infallibility(others) and contempt(from self)…………………….131

June 1983

5.1.(1)The child and the good god

The mathematical notes on which I am working now are the first since thirteen years that I destine to a publication. The reader will not be amazed that after a long silence, my style of expression has changed. This change of expression isn’t however the sign of a change in the style or in the method of work(1), and even less that of a transformation which would be made in nature itself of my mathematical work. Not only this has remained the same but I have acquired the conviction that the nature of the work of discovery is the same from one person who discovers to the other, that it is beyond the differences that the conditions and the temperaments varying to the infinity create.

Discovery is the privilege of the child. This is of the small child that I want to speak, the child who has no fear of yet being wrong, of looking like an idiot, of not being serious, of not doing like everyone else. He has no fear either of the things that he regards as having bad taste of being different from what he expects of them, of what they should be, or rather: of what, of course, they are. He ignores the silent and flawless consensus and without flaws which make part of the air that we are breathing - the one of all the proper people and well known as such. God knows that there have been many of them, of proper people and well known as such, since the night of the ages!

Our spirits are saturated from a heteroclite “knowledge”, a tangle of fears and laziness, of cravings and interdicts; of informations all comers and explanations push-button closed space where come to pile up informations; cravings and fears without ever engulfing the wind of the broad. Exception made of how to make a knowledge routine, it would seem that the principal role of this “knowledge” is of evacuating a living perception, a recognition of things from this world. Its effect is mainly the one of an immense inertia, a burden often crushing.

1 (1)

(Added in March 1984) It is without doubt abusive to say that my “style” and my “method” of work have not changed, whereas my style of expression in mathematics is transformed profoundly. The largest part of the dedicated time for a year to “The Prosecution of the Fields” has passed on my typewriter to write the reflections which are destined to be published virtually as is (in addition close to relatively short notes added subsequently in order to facilitate the reading by references, the corrections of errors, etc…). Neither scissors nor glue for preparing a laboriously “definitive” manuscript (which above all owes nothing to allow the path which are resulted to be transpired) - it makes nonetheless some changes of “style” and “method”! Unless to dissociate the mathematical work properly said from the written work, from presentation of the results, what is artificial, because that doesn’t correspond to the reality of things, the mathematical work being indissolubly linked to the writing.

The small child discovers the world as he breathes - the flow and the reflux of his respiration makes him welcome the world in his being delicate, and makes him in the world that welcomes him. The adult also discovers, in these rare moments where he has forgotten his fears and knowledge, when he regards things or himself with eyes wide open, eager to know, the new eyes - the eyes of child.

God has 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 has created the world and creates it day after day, in taking millions of millions of time, without respite, in groping, deceiving millions of millions of time and rectifying the fire, without to be tired…Every time, in this game of a lot of probe in things, of the response of things (“this is not bad this time there”, or: “there you are kidding in full”, where “it walks as on wheels, continues as that”), and of the new many rectifying probe or taking lots of previous probe, in response to the previous response…, in each go-and-return in this infinite dialogue between the Creature and the Things, which take place in each moment and in all places of the Creation, God learns, discovers, He takes knowledge of things more and more intimately, as they take life and form and being transformed between his hands. (as he takes?)

Such is the path of discovery and creation, such has it been for all eternity it seems (for as far as we are able to know). It was such, without man has had to make his entrance on the scene late, It has been scarcely one million of years or two, and that he puts the hand to the dough - with, recently, the unfortunate consequences that one knows.

It happens that one or the other of us discovers such thing, or such other. Sometimes he rediscovers then in his own life, with wonder, what this is to discover. Each has in him all that is necessary for discovering all which attract him in this vast world, including this wonderful capacity which is in him - the simplest thing, the most evident of the world! (One thing however that a lot have forgotten, as we forgot to sing, or to breathe as a child breathes…)

Each can rediscover what this that is discovery and creation, and no person can invent it. They have been there before us, and are what they are.

5.2. (2) Error and discovery

Going back to the style of my mathematical work properly said, or to its “nature” or to its “path”, they are now like before those that the good God himself has taught us without words to each, God knows when, very longtime before our birth perhaps. I do as him. This is also what each does instinctively, as soon as that the curiosity pushes him to know such thing among all, one thing invested from then by this desire, this thirst…

When I am curious of one thing, mathematics or other, I interrogate it. I interrogate it, without worrying myself if my question is stupid perhaps or if it is going to appear as such, without that it being at all price maturely weighed. Often the question takes the form of an affirmation - an affirmation that, in truth, is a blow of probe(query?). I believe there more or less, in my affirmation, it depends quite sure to the point where I am in comprehension of things that I am in the process of regarding. Often, above all at the beginning of a research, the affirmation is clearly false - again it is necessary to do it for being able to be convinced. Often, it suffices of writing it for it to jump to the eyes that this is false, so that before writing it there was a blurring, as a discomfort, in the place of this evidence. It permits now to return to the burden with this ignorance in minus, with one question - affirmation perhaps a little less “next to the plaque”. More often again, the affirmation takes from the foot of the letter proves false, but the intuition which, awkwardly again, has tried from expressing itself through it is just, all in remaining unclear. This intuition is gradually going to settle itself from a dross just as unformed first of false or inadequate ideas, it is going to exit gradually from the limbo of the misunderstanding that request only to be understood, from the unknown that request only to be left to know, for taking one form which only to it, to refine and brighten its contours, as the questions that I pose in these things before me more precise or pertinent, for surrounding them more and more closely.

But it happens also that by this path, the repeated blows of probe convergent towards one certain image of the situation, going out of the mists with some rather marked features for leading to a beginning of conviction that this image-there expresses the reality well - so that it is nothing however, when this image is tainted of an error of size, of nature to falsify it profoundly. The work, sometimes laborious; which leads to the route of one such false idea, from the first “take-offs” noticed between the image obtained and some patent facts, or between this image and others that had equally our trust - this work is often marked by an increasing tension, as one approches the node of the contradiction, that first makes itself from wave(not recognizing) more and more glaring - until the moment where it bursts at last, with the discovery of the error and the collapse of a certain vision of things, occurring as a immense relief, as a liberation. The discovery of the error is one of the crucial moments, one creative moment among all, in all work of discovery, whether it is a matter of a mathematical work, or of a work of discovery of self. This is a moment where our knowledge of probed thing suddenly renews itself.

To fear the error and fear the truth is one only and same thing. The one that fears of being mistaken is powerless in discovering. It is when we fear of deceiving ourselves that the error which is in us makes itself immutable as a rock. Because in our fear, we hang us to what we have decreed “true” one day, or to what since always has been presented to us as such. When we are driven, not by the fear of seeing an illusory safety disappear, but by a thirst of knowing, so the error, as suffering or sadness, crosses us without freezing ever, and the trace of its passage is a renewed knowledge.

5.3. (3) The unmentionable labors

This surely isn’t chance that the spontaneous path of all real research never appears for thus to say in the texts or the speeches which are supposed to communicate and transmit the substance of what has been “found”. Texts and speeches the most often to record limit themselves some “results”, under a form that to the common mortals has to show them as much of austere and unchangeable laws, included all eternity in the granite tables of a sort of giant library, and dictated by some omniscient God to the initiates-scribes-savants and assimilated; to those write the scholarly books and articles no less scholarly, who transmit knowledge from a pulpit, or in the more narrow circle of a seminar. Is there a single class of book, a single manual in the usage of the schoolchildren, high school students, students, or even “our researchers”, which is able to give to the unfortunate reader the least idea of what research is- if this isn’t justly the received idea worldwide that research, this is when one is very clever, that one has passed many exams and competitions as well, the large heads, Pasteur and Curie and the Nobel awards and all that…We other readers or audience, chowing as well as can be expected that these great men wanted to record for the good of the humanity, one is all just good (if one works hard) to pass our exams at the end of year, and again…

How much was there, including among the unfortunate “researchers” themselves, in bad thesis or articles, including even among the most “savants”, the most prestigious among us - who thus has the simplicity of seeing that “to do research”, this is neither more nor less that to question things, passionately - as a child who wants to know how he or his small sister came to the world. That to seek and to find, that is to say: to question and listen, is the simplest thing, the most spontaneous of the world, of which no one in the world has the privilege. This is a “gift” that we have all received from the cradle - made for expressing oneself and blooming oneself under an infinity of faces, from one moment to the other and from one person to the other…

When one risks hearing such things, one harvests among the ones as among the others, from the most stupid sure of being stupid, to the savant sure of being savant and well above the common mortals, the same smiles half-embarrassed, half-heard, as if one had just made a joke a little gross on the edges, as if one was with process of displaying naivete stitched of white wire; this is very beautiful all of these, must spit on anyone this is understood - but must not push even then - a dunce is a dunce and is neither Einstein nor Picasso!

Before an agreement as unanimous, I would have bad grace of insisting. Definitely incorrigible, I have again lost an opportunity to silence myself…

No, this surely isn’t chance if, with a perfect set, informative or uplifting books and manuals of all rests(not quite sure grabbed the right word) “the knowledge” as if it was out dressed from foot to head of the brilliant brains that has recorded for our benefice. One is not able to say either that this is from the bad faith, even in the rare cases where the other is quite “in the blow” for knowing that this image (that can not miss suggesting its text) matches in nothing to the reality. In one such case, it happens that the exposed presents more than a collection of results and recipes, that a breath crosses it, than a vibrant vision animates it, which sometimes then communicates from the author to the attentive reader. But a tacit consensus, of a considerable force it seems, makes that the text doesn’t allow the lesser trace to remain of the work of which it is the product, even when it expresses with a lapidary force the sometimes deep vision of the things which is one of the real fruits of this work.

As a matter of fact, in some moments I have myself confusedly felt the burden of this force, of this silent consensus, in the opportunity of my project of writing and publishing these “Mathematical Reflections”. If I try to probe the tacit form that this consensus takes, or rather that which takes the resistance in me to my project, released by this consensus, came to me immediately the term “indecency”. The consensus, internalized in me I would not know how to say since when, says to me (and this is the first time that I take the pains of draw to the light of the day, in the field of my view, what it mumbles to me a certain insistance for some weeks, if not some months): “It is indecent to spread before others, indeed publicly, the highest and the lowest, the useless trial on the edges, the “dirty laundry” in short, of a work of discovery. It doesn’t cause the time of the reader to be wasted, which is precious. Furthermore, it is going to make some pages and more pages, that it will be necessary to compose, print - what a mess, in the price where is the scientific printed paper! It is necessary to truly be vain for spreading like that things which have no interest for anyone, as if my messes themselves were remarkable things - an opportunity of strutting about, in sum”. And more secretly again: “It is indecent to publish the notes of such a reflection, such that it continues truly, just as it will be indecent to make love in a public place, or to expose or allow to be dragged, the sheets stained with blood from the labors of a delivery…”.

The taboo here takes the insidious and imperious form at the same time, from the sexual taboo. It is in the moment of writing this introduction that I begin to make out only its extraordinary force, and the scope of this fact itself extraordinary, attesting this force: that the true path of the discovery, from a simplicity so disconcerting, a infantile simplicity, shines through virtually nowhere; that it is silently retracted, ignored, denied. It is thus the same in the relatively harmless field of scientific discovery, not that of its dick or anything of such thank God - a “discovery” in sum good to be put among all the hands, and that (one would be able to believe) has nothing to hide…

If I would like to follow the “trace” which is introduced there, a wire nothing tenuous but all that there is a heavy and strong- surely it would lead me much further than a few hundreds of pages of algebra homological-homotopical that I finish well by terminating and delivering to the printer.

5.4. (4) Infallibility (to the others) and contempt (of self)

Definitely this was a euphemism, when sometimes I noticed carefully that “my style of expression” had changed, leaving even hearing that there was nothing there which can surprise: you understand well, when one hasn’t written for thirteen years, this is more similar than before, the “style of expression” it has to change, necessarily…The difference, this is that before I “expressed me” (sic) as anyone in the world: I did the work, I am able to do it again to the other sides, in erasing carefully every deletions. Path making, new deletions, turning upside down all the work sometimes worse than at the time of the rough outline. Therefore again - sometimes three times, indeed four, up to that everything is impeccable. Not only no doubtful corner nor pushed sweepings surreptitiously under a suitable furniture (I have never loved the sweepings in the corner, from the moment that one takes the pain of sweeping); but above all, in reading the final text, the certainly flattering impression that comes emitting itself from it (as any other scientific text) this is that the author (my modest person in this occurrence) was the infallibilityincarnate. Without fail, it falls directly on “the” good notions, then on “the” good statements, chaining itself in a purring of engine oiled well, with demonstrations that “fell” with a dull noise, each exactly in its moment!

That one judge of the produced effect on a reader who suspects himself nothing, a pupil of secondary school says learning the theorem of Pythagore or the equations of the second degree, indeed one of my colleagues of the institutions of research or education called “superior” (a word to the wise!) fencing himself (say) in the reading of such article of such prestigious colleague! This kind of experience is repeating itself of hundreds, of thousands of times all along a life of schoolboy, indeed of student or of researcher, amplified by the suitable concert in the family as in all of the media from all countries of the world, the effect is the one that one can provide. One finds it in oneself as in the others, for little that one gives himself the effort of there is attentive: this is the intimate conviction of his own nullity, by contrast with the skill and the importance of folks “who know” and folks “who make”.

This intimate conviction is compensated sometimes, but nothing resolved nor defused, by the development of a capacity in memorizing misunderstood things, even by the one of a certain operative entitlement: to multiply matrices, “mount” one French composition to blows of “thesis” and “antithesis”…This is the capacity in sum of the parrot or monkey savant, more popular of our days that it ever was, sanctioned by coveted diplomas, rewarded by comfortable careers. But the one there even stitched of diplomas and well fitted, covered of honors perhaps, isn’t dupe, at the very bottom of himself, of these artificial signs of an importance, of a “value”. Not even the one, more rare, who has invested his all on the development of some true gift, and who knew how to give his measure and make creative artwork in his professional life - he isn’t convinced, at the bottom of himself, by the brightness of his notoriety, by what often he wants to give exchange to himself and to the others. A same doubt ever examined lives the one and the other like the first came dunce, a same conviction of which ever perhaps they dare not to take knowledge.

This is this doubt, this unexpressed intimate conviction, which pushes one and the other to excel themselves without this in the accumulation of the honors or artworks, and to plan on others (on this before everything on which they have few power…) this contempt of themselves that gnaws in secret - in an impossible attempt to escape, by the accumulation of the “evidence” of their superiority on others(2)

2 (2)

(Added in March 1984) In rereading these last two paragraphs, I have had a certain feeling of discomfort, due to the fact that in the writing, I involved others but not myself. Visibly, the thought that my own person would not be able to be concerned touched me in writing. I have surely learned nothing, when I follow me narrow-minded well to put black on white (without doubt with a certain satisfaction) things that for years I have perceived in others, and eyesights to be confirmed from good ways. In the rest of the reflection, I leaded to me memory that attitudes of contempts face to face from others haven’t been lacking in my life. It would be strange that the link that I grabbed among contempt from others and myself is absent in the case of my person; the wholesome reason (and as the experience of similar situations of blindness to my own respect, of which I have finished by realizing me) says to me that it shan’t surely be so! This isn’t that however, for the moment, that a simple deduction, of which the only possible utility would be from encouraging me to see from my eyes what takes place, and see and examine (if it exists beautiful and good, or existed) this contempt of myself again hypothetical, id profoundly buried that it has totally escaped up to present in my view. It is true that the things to regard haven’t been lacking! This seems to me sudden as the most crucial one, from the fact exactly that it is in such hidden point…[(August 1984) To see however to this subject the reflection of the last two paragraphs of the note “The massacre”, n87.].

6. The dream and the dreamer

Contents

6.1. (5)The forbidden dream............................................. 133

6.2. (6)The dreamer............................................... 135

6.3. (7)The inheritance of Galois .......................................... 137

6.4. (8)Dream and demonstration......................................... 138

6.5. (9)The welcomed foreigner.......................................... 139

6.6. (10)The “mathematical Community”: fiction and reality....................... 142

6.7. (11)Encounter with Claude Chevalley, or: liberty and good feelings. . . . . . . . . . . . . . . . 143

6.8. (12)The merit and the contempt........................................ 145

6.9. (13)Strength and thickness........................................... 147

6.10.(14)Birth of the fear........................................ 150

6.11.(15)Harvesting and sowing......................................... 152

February 1984

6.1. (5) The forbidden dream

I take the occasion of an interruption of three months in the writing of the Pursuit of the Fields, for recapturing the Introduction at the point where I dropped it last June. I just reread it attentively, over six months of distance, and to added several subtitles from there.

In writing this Introduction, I stay well conscious that this type of reflections could not miss creating numerous “misunderstandings” - and it would be vain to try to get ahead of them, what would simply come back in accumulating from others on top of the first one! The only thing that I would add relative to this, is that it is by no means in my intentions of going to war against the style of scientific writing consecrated by a thousand-year-old usage, that I have myself practiced with constancy during more than twenty years of my life, and taught to my students as an essential part of the job of mathematician. Rightly or wrongly, today still I respect it as such and continue to teach it. Certainly even I would rather make old game, with my insistance on a work made to the end, sewed by hand from the beginning to the end, and without making thanks to any corner a little dark. If I had to put the water in my wine for ten years, it is well because of the strength of the things! The “writing in form” remains for me an important stage of the mathematical work, as much an instrument of discovery, for testing and going further into a comprehension of the things that remain without it approximative and sketchy, than as a means for conveying for such a comprehension. In the didactic point of eyesight, the mode of exposition of choice, the deductive mode thus, that excludes not at all the possibility of painting of vast pictures, offers some evident advantages, of concision and of convenience of the references. These are well there of the real advantages, and of weight, when it is a matter of expositions that address some mathematicians say, and more particularly, to some mathematicians who are already sufficiently familiar with certain sessions and meetings of the treated subject, or of all others nearby.

These advantages on the other hand become entirely illusory for a paper written to expose which is addressed to children, to young people or to adults who aren’t absolutely “in the blow” of advance, of which the interest isn’t already in awakening, and who however, the most often, are (and future, and with good reason…) in a total ignorance of what is the real step of a work of discovery. Some readers, for better to say, who ignore the existence itself of a such work, at the reach of each endowed with curiosity and good sense - this work of which bear and rebear without ceasing our intellectual knowledge of the things of the Universe, including what express itself in imposing arrangements as the “Elements” of Euclide, or “The Origin of the Species” of Darwin. The complete ignorance of the existence and of the nature of one such work is almost universal thing, including among the people at all the levels of education, from the primary school teacher to the professor of university. This is an extraordinary fact there, which to me is appeared in full light at the occasion first of the reflection begin the last year with the first part of this Introduction, at the same time that I made out then the profond roots of this disconcerting fact…

Then same that it would address to some readers perfectly “in the blow” to all points of view, there is one important thing left yet that the mode of exposition “of rigor” abstains from communicating. This is also one absolutely evil thing seen in the middle of serious people, as our scientific others in particular! I want to speak of the dream. Of the dream, and of the visions that it blows us - impalpable as it first is, and reticent often to take form. Of long years, indeed an entire life of intense work perhaps will not be enough to see to completely express themselves such vision of dream, to see to condense themselves and to be polished to it as far as the hardness and the brightness of the diamond. This is our work there, workers by hand or by spirit. When the work is achieved, or all part of the work, we introduce from there the tangible result under the most lively light that we are able to find, we rejoice us from there, and often pull pride from there. This isn’t in this diamond yet, that we have trimmed for a long time, that what find themselves to us has inspired in cutting it. Perhaps we shaped a tool of great precision, an efficient tool - but the same tool is limited, as all thing made by hand of the man, even when it shows us big. A vision, without name and without outline first, fine as a scrap of mists, guided our hand and kept us tending over the work, without feeling to pass the hours nor perhaps the years. A scrap which was detached without sound from a Sea without bottom of mist and of half-light…What is without limits in us is this, this Sea ready to rise to conceive and to give birth without ceasing, when our thirst fertilizes it. From this espoused there seeps out the Dream, such the embryo nested in the sustaining matrix, awaiting the obscure labors which will take it toward a second birth, to the light of the day.

Unhappiness to a world where the dream is despised - this is a world also where what is profond in us is despised. I don’t know if other cultures before ours - these of the television, of the computers and of the transcontinental rockets - have professed this contempt there. That has to be one of the numerous points through which we distinguish ourselves from our predecessors, that we have supplanted so radically, eliminated so to say from the surface of the planet. I haven’t had knowledge of another culture, where the dream is not respected, where these profond roots aren’t felt through all and recognized. And is there work of scope in the life of a person or a student, which isn’t born from the dream and wasn’t sustained by the dream before hatching to the daylight? With us yet (is it necessary even to say already: everywhere?) the respect of the dream is called “superstition”, and it is well known that our psychologists and psychiatrists have taken the measurement of the dream in long, in large and across - scarcely of what to clutter the memory of a small computer, surely. It is true also that no one “of us” knows how to start a fire, nor to dare to see his child to be born in his house, or to die his mother or his father - there are some clinics and hospitals that are there for this. God mercy…Our world, so proud of its power in atomic megatons and in quantity of information stocked in its libraries and in its computers, is without doubt the one also where the helplessness of each one, this fear and this contempt before the simple and essential things of the life has attained its highest point.

Fortunately the dream, all as the original urge of the sex in even the most repressive society, has a hard life! Superstition or not, it continues to conceal it from us to blow obstinately a knowledge that our awake spirit is too heavy, or too pusillanimous to apprehend, and to give life and to lend some wings to the projects that it inspired us.

If I allowed to be heard sometimes that the dream was often reticent to take form, it is a matter there of an appearance, which doesn’t truly touch to the bottom of the things. The “reticence” would rather come from our spirit of the state of yesterday, in its ordinary “plate” - and again the term “reticence” is euphemism to it! It is rather a matter of a profond mistrust, which recovers an ancestral fear - the fear of knowing. Speaking of the dream in the proper sense of the term, this fear is accordingly more acting, it makes an accordingly more efficient screen, than the message of the dream touches us of more close, that it is heavy of the threat of a profond transformation of our person, if by chance it was coming to be heard. But it is necessary to believe that this mistrust is present and efficient even in the relatively harmless case of the mathematical “dream”; to the point that every dream seems banished not only from the texts (I know none in any case where there was trace of them); but equally of the discussions between colleagues, in small committee, indeed in private.

If it is like this, this isn’t admittedly that the mathematical dream would not exist or not anymore - our science then would have became sterile, what is by no means the case, surely the reason of this apparent absence, of this conspiration of the silence, is tied very close to the other consensus - the one of erasing carefully all trace and all mention of the work through which is made the discovery and renews our knowledge of the world. Or rather, this is a single and same silence that surrounds both the dream, and the work that it creates, inspires and nourishes. To the point that the term even of “mathematical dream” will seem a nonsense, moved as we are so often by some push-button cliches, rather than by the direct experience that we are able to have from a reality completely simple, daily, important.

6.2. (6) The Dreamer

In fact, I know well by experience that when the sprit is eager of knowing it, instead of avoiding it (or of tackling it with a patented gate at hand, what come back in the same), the dream is by no means reticent “to take form” - to allow itself to describe with delicacy and to deliver its message, always simple, never silly, and sometimes deeply distressing. Well on the contrary, the Dreamer in us is an incomparable ruler for finding, or creating of all pieces, from one occasion to the other, the cleanest language to circumvent our fears, to shake our torpor, with some scenic means varying to infinity, since the absence of all visual or sensorial element what that is, to the most staggering scenes. When It(God) appears, this is by no means for slipping away, but for encouraging us (in pure loss almost today, without that Its(God’s) kindness grows weary…) to go out from ourselves, from the heaviness where it sees us restricted, and that it enjoys itself sometimes, face of nothing, from parodying in some funny colors. To lend ear to Dreamer in us, this is to communicate with ourselves, against the powerful dams which would like at all costs we forbid it.

But what is the most able to, is the least able to. If we are able to communicate with ourselves through the intervention of the dream, showing us to ourselves, surely it must be possible in a way as simple to communicate to others the message by no means intimate of the mathematic dream, say, which doesn’t put in play of the forces of strength of a comparable power. And to tell the truth, what have I done else in my past of mathematician, if it isn’t following, “to dream” as far as the end, as far as their the most obvious expression, the most solid: unimpeachable, of the scraps of dream coming away one by one from a heavy and dense fabric of mists? And how many times have I stamped from impatience before my own obstination to polish jealously as far as its last every precious stone facet or precious to half in what was condensing my dreams - rather than of following the many-sided mysteries of the fabric-mother - on the indecisive borders of dream and of its patent incarnation, “publishable” in sum, following the canons in vigor! I was moreover on the point of following this impulsion there, of throwing myself in a work of “mathematic science fiction”, “a sort of awake dream” on a theory of the “motifs” which was remaining to this purely hypothetical moment - and which is remained it until today still and for cause, lacking to another “awake dream” to launch into this aventure. It was towards the end of the 60’s, at that time that my life (without that I doubt the least of the world) was getting ready to take a whole other curve, which during ten years was going to relegate my mathematic passion to a marginal place, or even repudiated.

But to take everything well, “To the Pursuit of the Fields”, this first publication after fourteen years of silence, is well in the spirit of this “awake dream” which was never written, and of which it seems to have followed the provisional result. Admittedly, the themes of these two dreams there are as dissimilar, to first sight at the very least, that it is possible for two mathematic themes; without counting that the first, the one of the motifs, would seem to be situated at the horizon rather of what would be able to be “feasible” with the means of the edge, while the second, the famous “fields” and company, seems absolutely within reach of the hand. These are some dissimilarities there that one would be able to call fortuitous or accidental, and that perhaps will vanish much earlier than one expects it(3). They only relatively few of incidence, it seems to me, on the type of work which one and the other theme are able to give rise, from the time of that it is exactly a matter of “awake dream”, or, for saying it in less provocative terms: of chasing the work of rough conceptual as far as a whole vision and of a sufficient precision, for pulling the conviction more or less complete that the vision corresponds well, for the essential, to the reality of the things. In the case of the theme developed in the present work, it would have to mean, more or less, that the detailed verification of the validity of this vision becomes a question of pure skill. It is able to admittedly demand a considerable work, with its part of trick and of imagination, and without doubt as of the new development and of the unexpected perspectives, which will make in other thing, fortunately, that a work of pure routine (a “long exercice”, as Andre Weil would say).

This is the type of work, in sum, that I did and did again to my fill in the past, that I have at the tip of my fingers and therefore it is useless that I do again in the years which remain still ahead of me. In the measure where I invest myself again in a mathematic work, it is on the borders of “awake dream” that my energy will surely be the best used. In this choice, this isn’t of elsewhere a problem of profitability which inspires me (to suppose that such a problem is able to inspire whoever), but a dream exactly, or some dreams. If this new momentum in me has to be revealed as bringer of strength, it is in the dream that it will have been drawn!

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I think here in particular of the previous conjectures of Mordell, of Tate, of Chafarevitch, which found themselves demonstrated all last three years in a manuscript of forty pages of Faltings, at a moment where the well established consensus of the people “in the blow” ruled these conjectures were “out of reach”! It is found that “the” fondamental conjecture which serves of key to the vault of the programme of “non-abelian algebraic geometry” which is dear to me, is exactly close to the conjecture of Mordell. (It would appear same that this one would be a consequence of that one, which showed well that this programme was not a story for serious people…)

6.3. (7) The heritage of Galois

It would seem that among all the natural sciences, this is only in mathematics that what I called “the dream”, or “the awake dream”, is stricken of a apparently absolute prohibition, twice more than a thousand times. In the other sciences, there included of the “exact” renowned sciences as physics, the dream is for at least tolerated, encouraged indeed (according to the ages), under names it is true more “acceptable” as : “speculations”, “hypothesis” (such the famous “atomic hypothesis”, coming from a dream, pardon of a speculation of Democratic), “theories”… The passage of the status of the dream-that-not dare-to say-its-name to the one of “scientific truth” makes itself by insensible degrees, by a consensus that widens progressively. In mathematics on the other hand, it is a matter almost always (of our days at least) of a sudden transformation, by the virtue of the wave of magic wand of a demonstration(4). In the times where the notion of mathematical definition and of demonstration wasn’t, as today, clear and consensus (more or less) in general, there was however some visibly important notions that had an ambiguous existence - as this of “negative” number (rejected by Pascal) or this of “imaginary” number. This ambiguousness is reflected in the language in usage still today.

The progressive clarification of the notions of definition, of terms, of demonstration, of mathematical theory, has been in this regard very beneficial. It caused us to take consciousness of all the power of the tools, of a children’s simplicity yet, of which we make use for formulating with a complete precision even that which could seem informulable by the single virtue of a sufficiently rigorous usage of the common language, tofew of things nearby. If there is one thing that fascinated me in mathematics since my childhood, this is exactly this power to surround by the words, and to express of completed way, the essence of these mathematical things which at the first introduce themselves under a so elusive form, or so mysterious, that they seem beyond words…

A psychological yet unfortunate repercussion of this power, of the ressources that completed precision and the demonstration offer, this is that they accented again the taboo traditional to the respect of the “mathematical dream”; this is to say to the respect of all what wouldn’t introduce itself under the conventional aspects of precision (being this at the expense of a more vast vision), guarantee “good dyed” by some demonstrations in form, or otherwise (and more and more by the times that run…) by some sketchs of demonstration, supposed to be able to put themselves in form. Some occasional conjectures are tolerated by necessity, to condition that they satisfy to the conditions of precision of a questionnaire, where the only permitted replies would be “yes” or “no”. (And also on the condition, it is necessary to say, the one who permits himself making it one who is respected in the mathematical world.) To my knowledge, there hasn’t been example of the development, called “experimental”, of a mathematical theory that would be explicitly conjectural in its essential parts. It is true that following the modern canons, all the calculations of the “infinite small” developed to go since the seventeenth century, became since the differential calculation and integral, would take figure of awake dream, which would itself be transformed finally in

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Even of our days moreover, one encounters of the “demonstrations” to the uncertain status. It has been so during years of the demonstration by Grauert of the theorem of finiteness which carries his name, that no one (and the good wills haven’t been lacking!) happened to read. This perplexity has been solved by other more transparent demonstrations, and of which some went farther, which have taken the succession of the initial demonstration. A similar situation, more extreme, is the “solution” of the problem called “of the four colors”, of which, the calculation part has been well ordered work of computer (and of several millions of dollars). It is a matter thus these of a “demonstration” that doesn’t find itself better-founded in the intimate conviction coming from the comprehension of a mathematical situation, but in the credit that one makes to a machine devoid of the power of understanding, and of which the mathematician user ignores the structure and the working. To suppose even that the calculation is confirmed by the other computers, following other programmes of calculation, I don’t consider for as much that the problem of the four colors is closed. It will have only changed face, in this sense that it is scarcely any longer a matter of finding a counter-example, but simply a demonstration (readable, it goes of itself!).

serious mathematics two centuries later only, by the blow of magic wand of Cauchy. And it put me back in mind of necessarily of the awake dream of Evariste Galois, who hasn’t had chance(luck) with this same as Cauchy; but it was enough this time at least for a hundred years for that another waving the wand, of Jordan this time (if I remember well), gives roads of the city to this dream, renamed for the circumstance “theory of Galois”.

The constatation which emits itself of that, and which isn’t to the advantage of the “1984 mathematics”, is that it is happy that some people like Newton, Leibnitz, Galois (and I pass surely a lot, not being clever in history…) haven’t been encountered by our present canons, in a time where they were contenting themselves of discovering without taking the leisure time of canonizing!

The example of Galois, came there without that I call it, touches a sensible chord in me. It seems to me to remind that a feeling of fraternal sympathy have missing to his strange destiny, to the time where I was still a high school student or student, I believe. Like him, I felt in me a passion for the mathematics - and like him I felt to me a dropout, a stranger in the “beautiful world” that (it seemed to me) had rejected him. I finished however myself by making part of this beautiful world, for leaving it one day, without regret… This affinity a little forgotten is reappeared to me very recently and under a day all new, when I wrote the “Sketch of a Programme” (on the occasion of my demand of admission as researcher to the Center National of the Scientific Research). This report is consecrated principally to a sketch of my principal themes of reflection for ten years. Of all these themes, the one that fascinates me the most, and that I count to develop above all in the next years, is the type even of a mathematical dream, which joins elsewhere the “dream of the motifs”, of which it supplies a new approach. In writing this Sketch, I remembered the longest mathematical reflection that I pursued of a draft in these last fourteen years. It was followed from January to June 1981, and I named it “The long Walking through the theory of Galois”. One thing leading to another, I became awarer that the awake dream that I chased sporadically for several years, which finished by taking the name of “non-abelian algebraic geometry”, was other only a continuation, “a final successful realization of the theory of Galois, and in the spirit without doubt of Galois”.

When this continuity appeared to me, at the moment of writing the passage of which is extracted the quoted line, a joy crossed me, which hasn’t dissipated itself. It has been one of the rewards of a work pursued in a complete solitude. Its apparition has been as unexpected as the more welcome than cool received formerly from two or three colleagues and former friends yet well “in the blow”, of which the one however was my student, to which I had had the occasion of speaking, “to warm” again and in the joy of my heart, of these things that I was in the process of discovering…

It reminds me that to recapture today the inheritance of Galois, this is surely as to accept the risk of the solitude which was his in his time. Perhaps the times they change less than we think, often this “risk” does not take for me the appearance of threat. If it happens to me of being struggled and frustrated by the affectation of indifference or of disdain of those that I loved, never, on the contrary, as long years of solitude, mathematics or other, it has weighed on me. If ever there is a faithful friend that I always aspire to find again when I have just left it, it is the solitude!

6.4. (8) Dream and demonstration

But come back to the dream, and to the prohibition that hit it in mathematics for thousands of years. This is perhaps the most ingrained there among all the preconceptions, often implicit and deep-rooted in the habits, decreeing that such thing “this is of the maths” and such other, no. These had to be thousands of years before the things as childlike and omnipresent as the symmetry groups of certain geometric figures, from topological forms of certain others, the number zero, the admissible sets in the the sanctuary! When I speak to the students about the topology of a sphere, and of the forms that deduct themselves of a sphere, adding of the coves - things that don’t surprise the young children, but which divert them because they believe to know what is “of the maths” - the first spontaneous echo that I receive is: but this is not math, certainly, this is the theorem of Pythagore, the heights of a triangle and the polynomials of the second degree… These students are not more stupid than you or me, they react as have reacted from all time until today all the mathematicians of the world, except people like Pythagore or Riemann and perhaps five or six others. Poincare himself, who did’t come the first, happened to prove through an philosophic A plus B well-chosen that the infinite sets, this was not maths! Surely there must have been a time where the triangles and the squares was not maths - it was of the drawing that the kids or the artisan potters drew on the sand or in the clay of the vases, not to confuse…

This basic inertia of the spirit, suffocated by its “knowledge”, isn’t only admittedly to the mathematicians. I am in the process of moving myself away a little from my proposal: the prohibition that hits the mathematical dream, and through it, all what doesn’t present itself under the usual aspects of the finished product, ready to the consumption. The little that I have learned on the other natural sciences is enough to make me to measure that a prohibition of a similar rigor would have condemned them to sterility, or to a progress of tortoise, a little like in the Middle Ages where it wasn’t question of (?) the letter of the Holy Writings. But I know well also that the profound source of the discovery, must like the step of the discovery in all its essential aspects, is the same in mathematics as in every other region or thing of the univers that our bodys and our spirit are able to know. To banish the dream, this is to banish the source - to condemn it to a secret existence.

And I know well also, by an experience that didn’t deny itself since my first and young loves with the mathematics, this: in the deployment of a vast or profondvision of the mathematical things, it is this deployment of a vision and of a comprehension, this progressive penetration, which constantly precedes the demonstration, that renders it possible and gives to it its sense. When a situation, from the most humble to the most vast, has been understand in its essential aspects, the demonstration of what is understood (and of the rest) falls as a completely ripe fruit. Then that the demonstration pulled out as a fruit still green on the tree of knowledge leaves a back-taste of insatisfaction, a frustration of our thirst, by no means calmed. Two or three times in my life of mathematician I had to resolve, for want of anything better, to pull out the fruit rather that to pick it. I don’t say that I have done badly, or that I regret it. But what I have known to do better and what I have loved the most, I have taken it willingly and not by force. If the mathematics has given me joys to profusion and continues to fascinate me in my mature age, it isn’t by the demonstrations that I would have known to pull it out, but by the inexhaustible mystery and the complete harmony that I feel in it, always ready to be revealed to a hand and a magnet look.