The title " The Funeral Rites" has been given to all of the footnotes relating to " The Weight of the Past". It's suitability came to me with insistent force in the course of writing this reflection (*)
(*)Near the end of this work another name also suggested itself, expressing a different, equally striking aspect of a certain picture of things that revealed itself to me in progressive stages over a period of 5 weeks. Its the title of a fable, to which I will return in its proper place: The Robe of the Emperor of China
In it I play the role of a man whose death is anticipated, in the lugubrious company of several mathematicians ( much younger than I), whose work has all been done after my "departure" in 1970, who therefore lacked the advantages of direct contact with me and my advice, who knew about my work through my writings, those published or available by other means. At that time I was already being treated as something of a corpse, to the degree that for a very long time that even the notion that people ought to meet me was not current, and that a continuous relationship ( as much personal as mathematical), had unravelled about a year before.

That did not however prevent Mebkhout, going against the grain of a somewhat tyrannical and disdainful attitude of his peers ( who were my students) , and in nearly total isolation, to create original and profound work through making a unexpected synthesis of my ideas with those of the school of Sato. His work opened a new insight into the cohomology of analytic and algebraic varieties; it carries the promise of a vigorous renaissance in our understanding of this cohomology. It is certain that this resurgence of research in this subject would have been carried through by now, or even several years ago, had Mebkhout found himself surrounded by the appropriate warmth and unreserved support from his colleagues, which he had formerly received from me. As it is, since October 1980 his works and ideas have supplied the inspiration and the technical means for spectacular advances in the theory of the cohomology of algebraic varieties, which has finally emerged ( putting aside the results of Pierre Deligne on the Weil conjectures), after a long period of stagnation.

Incredible as it may sound, over the last four years his ideas and results have been used by "everybody" ( just as mine have been), even as his name is ruthlessly ignored and suppressed, even by those who know about his work through direct association with him or who have used it as an essentially component of their own research. I know of no other period in the history of mathematics which has been guilty of such disgraceful conduct, namely, that some of the most prestigious and influential members of the community set the example for all others, to violate the most fundamental ethical principles of the mathematical calling.

I distinguish four men, all brilliant mathematicians, who have shared with me the honor to be the victims of this burial through silence and disdain . I can see how the stigma of contempt has poisoned, in each of them, the beautiful passion for mathematics that once inspired them.

Quite distinct from them I see above all two men, each of them a monument set up in places of public mathematical honor, who are now acting in official capacities in the funereal rituals, and who, at the same time, ( in a more secretive sense ) are burying themselves with their own hands. One of these has already been named. The other is also one of my former friends and students : Jean-Louis Verdier . Apart from occasional and brief meetings at professional conferencesm, We did not keep up contact after my "departure" of 1970. No doubt this explains why he does not figure overly much in this reflection apart from certain of his professional activities. Furthermore the motivation behind these activities, vis-a-vis his relationship to me, are not looked into at all because I don't know anything about them

If there is one inquiry which has, with great urgency, driven me over the long years, which served as a deep incentive for writing Récoltes et Semailles, which has accompanied me throughout the length of its writing, it is that of my own responsibility in the advent of a certain kind of spirit and a certain kind of tradition which have made possible the kinds of demeaning behavior to which I've alluded, from a world to which I belonged and to which I entirely identified over the more than twenty years that I was a mathematician. This self-examination has led to the discovery of distinctively fatuous attitudes in me, which manifested themselves by a disdain for colleagues less gifted than I was, and by a spirit of accomodation to mathematicians at my level. I therefore am no stranger to the kinds of attitudes which I see everywhere around me today, among persons I've loved, and among those to whom I taught a subject that I loved. One can say that it is among those whom I've badly loved and badly instructed who are setting the tone ( let us not call it the law), in a world that was so dear to me, and which I finally left.

I sense an atmosphere of self-congratulation, of cynicism and contempt. "The wind bloweth where it listeth..." ... I have understood that we are reaping the whirlwind of all those blind and callous seeds which I helped to plant. And if this whirlwind has fallen back onto me, and on those whom I've left in other hands, and on those for whom I still have affection and who have had the courage to admit that I inspired them, it is not more than a tit-for-tat of which I have no-one to blame but myself, and which has much to teach me.

7.In the table of contents, under the designation of "The Funeral Rites" , I have therefore collected the impressive lost of the principal "notes" relating to that seemingly innocuous section " The Weight of the Past". These notes give meaning to the name which impressed itself upon me for this final section of the first draft of Récoltes et Semailles.

In this long processional of footnotes of multiple parentage, including those which have been added over the last 4 weeks ( notes (51) to (97) are the only ones which have dates ( from April 19th to May 24th(**)

(*) Note #104 of May 12 1984 has also be added. The notes from #98 onwards ( with the exception of the footnote that precedes #104 ), form the "third wind" of this reflection, which began on September 22, 1984. All of them are dated . In a list of footnotes which had been written up on the same day, only the first is dated. Others among them which aren't dated are #44' to #50 ( these form the funeral processions I, II,III) . Footnotes #'s 46, 47, and 50 are from March 30th or 31st, footnotes #44', 48, 48', 48'', 49 from the first half of April. Finally, note #44" is dated May 10th.
(**) From time to time I've weakly inverted the chronological order, to the benefit of the "logical order", whenever I felt that it was important to maintain the sense of a progression advance in my reflections. The only exceptions are the 11 footnotes ( whose numeration is always preceded by the exclamation sign (!) ). They refer to other footnotes which risked swelling to prohibitive dimensions. They've been placed directly after the footnotes to which they correspond. ( with the sole exception of note #98, which actually relates to note #47.)
(***) When the numeral of footnote ( such as (46) ) is referred to within a note in the section entitled " The Weight of the Past", then the numeral of the note containing that reference ( such as (50) for example) is placed immediately after the numeral of the referenced footnotes . In this example that would be #46 (50) .
(****)The numbering of a footnote which is the direct continuation of the preceding one, is always preceded by an asterisk * in the table of contents. Thus for example, *47.46 indicates that note #47 is the immediate continuation of the subject matter of #46 ( which may well be different from the note that physically follows #46, which is this case is the footnote #46.9.Finally I have indicated, in the table of contents, the numbers of all the footnotes that are not the continuations of others, those which represent a "new departure" in my thinking, and have no clear place with regard to the previous reflection
In order to lend a bit of structure to the totally of The Funeral Rites, and so that one doesn't lose one's place in the multitudes of footnotes, it seemed a good idea to me, depending on the circumstances, to adjoin at certain places a number of very suggestive subtitles. They precede and orient a long succession of consecutive footnotes united around a common theme.

I've thereby has the signal satisfaction of witnessing the gradual coming together, piece by piece, of the 10 phalanxes of the long processional assembled for my burial rites (*)

(*) September 29th. In fact there are actually 12 phalanxes, including that of the hearse (X) , and "The corpse ( who is however not yet dead) " (XI) , which arrive in extremis to insinuate themselves into the processional.
Some of them modest, others imposing, some contrite, others secretly exulting, as it must always be on such occasions. One by one these advance

  1. The Posthumous Pupil (whom all consider it their duty never to acknowledge)

  2. The Orphans (freshly exhumed for this solemn occasion

  3. The Illustrious Men in Fashion ( I deserved that one)

  4. The "Motives" ( the latest born and most recently resurrected of all my orphans)

  5. My friend Pierre D ... (the humble leader of the largest of all the phalanxes) , followed close behind by

  6. The Unanimous Concord of Silent Melodies

  7. The 'Colloquium" (alias 'The Perverse') , which includes a full house. For the posthumous pupil, otherwise known as the 'unknown pupil' , subsidiary funeral marches carrying flowers and crowns have been set aside.

    Finally, bringing up the rear of this imposing brigade, we watch the advancement of

  8. The Pupil ( hardly posthumous though even less known), alias The Boss, followed by

  9. The harried troupe of all my students ( forced to carry shovels and buckets) . Finally:

  10. The Hearse, ( holding in its keep four beautiful coffins of solid oak, their lids well screwed in place, plus the Gravedigger )

Altogether 10 phalanxes all coming together ( it was high time) to move, ever so slowly, to The Ceremony.

The final nail in the coffin will be the Funeral Oration, served up with the finest of touches by none other than my old friend Pierre in person, presiding over the rites at the request and satisfaction of all present. The Ceremony terminates with a (final and definitive; at least one hopes )De Profundis, chanted as a sincere act of contrition by none other than the much lamented corpse himself, who, to the stupefaction of all present has somehow survived his own obsequies and even participated in them with the greatest imaginable satisfaction - this satisfaction providing the final note and the final Tierce de Picardie(Note: it is the translator who finds this technical term from music particularly apt.) of so memorable an entombment.

8. In the course of this final stage ( so we hope) of this retrospective it was felt that an Appendix was needed for volume 1 of the "Mathematical Reflections", containing two other documents of a mathematical character, in addition to the three already alluded to (*)

(*) In addition I've considered adding a commentary to the "Thematic Outline", giving some more details about my contributions to the "themes" which are summarily passed over, as well as on the subject of the influences at work in the genesis of the major and most powerful ideas in my mathematical opus. An overview of the last six weeks has already led me to realize ( much to my surprise!), to what extent Jean-Pierre Serre played the role of a "detonator" to the eruption of most of these ideas, as he did for several of the "grand projects" which I envisaged between 1955 and 1970.

Finally, there is another mathematical text ( in the modern sense), the only one to figure in the body of Récoltes et Semailles, that I want to call attention to, which is the sub-footnote #87 adjoined to the footnote "The Massacre" ( #87)in which I've described, with great care, my conjecture of a discrete 'variant' of the familiar Riemann-Roch-Grothendieck theorem for the continuous context. This conjecture figures (along with several others) in the treatment at the close of the seminar SGA 5 of 1965/66, of which, ( along with other work) not a trace is to be found anywhere in the volume that was published 11 years later under the title "SGA 5" . The vicissitudes of this critical seminar at the hands of some of my students, and their ties to a certain Operation SGA 4 1/2 are unfolded bit by bit in the investigation carried out in the footnotes #63''', 67, 67', 68, 68', 84,85, 85', 86, 87, and 88.

Another mathematical discussion, about the possibility of putting together a topos ( to the extent possible) for the known cases in which there exists a formal duality of the sort I've called the "6 operations", is to be found in the subnote #81.2 of the footnote "Risk free insurance to a thesis advanced on credit", #81.

The first text reproduces and comments on a report in two parts which I did between 1968 and 1969 on the works of Pierre Deligne ( much of which remain unpublished to this day) , corresponding to the mathematical activity at the Institute des Hautes Etudes Scientifiques during the years 1965/67/68.

The other text is a sketch for " a formulation in six variances" , bringing together the common features of a duality formalism ( inspired by those of Poincaré and Serre) which I drew up between 1956 and 1963, a 'formula' which lays claim to being 'universal' for every situation of cohomological duality encountered up to the present day. This formulation appears to have fallen into oblivion after my departure from the mathematical world, to the extent that, to my knowledge and apart from myself, no one has bothered even to draw up a list of the fundamental operations, those fundamental canonical isomorphisms which they engender, and the resemblances between them.

This sketch for a coherent formulation would turn out to be for me the first step towards that "grand delineation of the dream of the motives " which, for more than 15 years, " awaits the bold mathematician who would dare to tackle it". It appears to be the case that this mathematician has to be me. Indeed it is high time that this notion, born in my private reflections over twenty years ago, which was never intended to be the property of a single person but was destined for all, should finally emerge from the obscurity of night, to be born once again in the full light of day.

It is the case of course that there's only one person, apart from myself, who has developed an intimate knowledge of this 'yoga of motives' , who in fact learned about it directly from me in the days and years preceding my departure. Among all the mathematical discoveries which I've been privileged to make, the concept of the motive still impresses me as the most fascinating, the most charged with mystery - indeed at the very heart of the profound identity of geometry and arithmetic . And the yoga of motives which brought me to this now much neglected reality is perhaps the most powerful research tool invented by me during the first period of my life as a mathematician.

Yet it is also true that this reality, and this "yoga" that closely surrounds it, were never kept as personal secrets. Absorbed as I was in the urgent task of communicating the fundamentals ( which since then everyone else is happily content to use in their daily work) , I could not find the extra months needed to edit an enormous sketch I'd drawn up about the yoga of motives in its totality, thereby putting it at the disposal of everyone. But I did not fail, in the years before my departure, to speak about them at conferences or to anyone who cared to listen to me, beginning with my students who ( with the exception of one of them), have totally forgotten everything I taught them, just like everyone else.

If I speak of them now, it is not with the intention of augmenting the list of 'inventions' that bear my name, but rather to draw attention on a mathematical reality which is virtually self-evident once one interests oneself in the cohomology of algebraic varieties, particularly of their arithmetical properties, and of their relationship to all other cohomological theories current at the present time. This reality is as concrete as, in the past, the notions of infinitely small entities were, which were understood and used long before a rigorous mathematical formalism officially established their legitimacy. And, to understand the reality of motives we are not in any short supply of a flexible language for describing them, nor do we, like our predecessors, lack experience in building mathematical theories.

Although all that I have shouted from the roof tops has, up to the present time, fallen on deaf ears, and although my disdainful elective mutism has received, as an echo, the silence and laziness of all those who claim to be interested in cohomology ( and who all the same have hands and eyes just like mine) , I cannot hold alone responsible that person who has chosen to guard for himself as a kind of personal treasure all that I'd confided in him that was intended to benefit everybody. One is however forced to the conclusion that our age, whose frenzied productivity in the domain of science is at a level with that of weapons and the consumption of material goods, is at the same time a long way from that sort of "bold dynamism" of our 17th century predecessors, who did not wait to receive support from the four corners of the earth before develop an infinitesimal calculus, or worry about whether what they were doing was rigorous or pure conjecture, or wait for some eminence among them to give them the green light.They were not afraid to grasp and work with that which everybody could see first-hand with their own eyes.

9. By virtue of its inner structure and the nature of its theme, the "Funeral Rites" (which now forms more than half of the text of Récoltes et Semailles is largely independent from the long reflection that precedes it. This independence is however only apparent. For myself this meditation on a "burial" that continues to emerge bit-by-bit from the fog of the unspoken, is inseparable from what comes before, from which it grew, and which gives it all its meaning. Begun as a quick glance, in passing, on the trials of a life's work which I'm grown somewhat out of touch with, it became, without any intention on my part, a meditation on one of the important relationships in my life, leading me in turn to another disquisition on the fate of this work in the hands of "those who were my students". To make a separation between this work and that from which it has spontaneously emerged seems to me to be a kind of simplistic reduction of this reflection to a kind of Bildungsroman (which might even be misinterpreted, in that wonderful world of mathematics, as a kind of 'settling of accounts')

It becomes such if one thinks of it in that way: a similar interpretation as a "novel of manners" could be made of the whole of Récoltes et Semailles. It is certainly the case that the customs prevailing in any given age, in a certain milieu, which shape the lives of the people who participate in it, are important and ought to be identified. However it should be clear to any careful reader of Récoltes et Semailles that I've not set out to describe manners or customs, that is to say to depict a certain kind of 'scene', one that changes with time and place, which serves as a backdrop for our acts. To some extent this backdrop defines and restricts the means at the disposal of whatever there is in us that we would wish to express. Although the setting and its opportunities ( as well as its "rules of the game") are infinitely varied, the forces deep within us that (at the collective level) shape the settings and which ( at the personal level), are expressed within these settings, seem to be essentially unvarying from one culture to another or from one age to the next. If there is one thing in my life, apart from mathematics and a love of women, that has evoked for me a sense of supreme mystery (quite late it's true) , it is indeed the hidden nature of these forces which determine our actions, for better or worse, for destruction or creation.

10. The reflection which eventually took the name of "The Funeral Rites" had been initiated as a token of respect : respect for the things I'd discovered, that I'd watched in the process of formation from the void, of which I was the first to savor and to appreciate the strength, to which I'd given names to express the way in which I came to know them, and, as already stated, to show my respect for them. These are the things to which I gave the best of myself, nourished by my interior force. They came out of the earth and blossomed forth like the many branches of the same tree trunk, bristling with many vigorous roots. They continue to live, not only as inventions that one might chose to make or not to make - but all closely connected and united in one vital whole that is formed from each and gives each its place and sense, its origin and its end. They were abandoned long ago, without anxiety or regret, because I knew that what I was leaving was healthy and strong and had no more need of me to flourish, to be fruitful and to multiply, following its proper nature. It is not a mere sack of gold coins that I leave, that anyone can steal, nor a bundle of tools that can rust and decay.

However, over the course of many years, during which I felt myself distanced from the world I'd left, there came to me in my retreat, faint suggestive indications of a spirit of insidious contempt and discrete derision attached to things that I knew to be strong and beautiful, having their proper place, functioning, and irreplaceable. I had the feeling that I'd left a family of orphans to cope alone against a hostile world, a world sick with the disease of contempt, attacking all things unable to fend for themselves. It was in this state of mind that I began this disquisition, as a token of respect towards these things and towards myself - responding to the pull of a deep bond between these things and myself. He who affects to denigrate any one of these things which have been nourished by my love, is in a sense denigrating me as well, and everything that had issued from me.

And the same thing may be said of someone who, knowing at first hand of the tie that unites me to certain things which he learned from none other than me, would pretend that this tie was of no significance, or to claim to be unaware of its existence, or to make claim (either overtly or through omission) on his account or that of someone else, of a fictive "paternity". These clearly represent, to my mind, acts of contempt for the products of my labor, as well as the obscure and subtle work involved in bringing these things to birth, and to me, the worker himself, and indeed, ( in a more hidden yet essential way), that individual himself.

If my "return to mathematics" will have no other effect than that of making me once more aware of that bond, and in bringing out this token of my respect to the awareness of others - those who affect contempt and those who pretend to indifference - it will not have been in vain.

It is true enough that I'd really lost contact with the works (written and unwritten, or at least unpublished) , that I'd left behind. When I started this reflection I saw all the branches distinctly, without remembering how they were all connected up to the same tree. Strangely enough, it took the gradual revelation of a spectacle of pillaging of what I'd left, for me to discover the vital unity of that which has been ruthless plundered and dispersed. One took the bag of gold coins, another a tool or two from the toolbox, to claim credit for himself or to exploit in some other way. Yet the unity which gave life and the true force behind what I'd left has totally escaped all of them.

Still, I know of a single person who was deeply touched by this unity and this force, and, in his inner depths still feels it, whom it pleases to disperse this force and who has it in him to destroy this unity which he discovered in another ( by the intermediary of his works) . It is in this living unity that one finds the beauty and the creative virtue of this opus. Notwithstanding the pillage, I find them intact, just as they were when I left, though I have matured and see them today with new eyes.

If I find that these things have been despoiled and mutilated, robbed of their initial force, I know that it has been done by those unaware of their own interior force, who think they can get away with despoiling at their pleasure. Yet all they are doing is cutting themselves off from the creative virtue which is at their disposal as it is at the disposal of everyone, yet never at the mercy or power of another.

Thus this reflection, and, through it the unexpected "return", has led me to reestablishing contact with a forgotten beauty. It is the sensation of this beauty which gives meaning to this act of respect, but poorly expressed in the note "My orphans"(*), which I invoke now, in full recognition of its significance.

(*) This note (#46) is chronologically the first to figure in "The Funeral Rites" .

Promenade through a life's work- or a child and his mother

1: "The magic of things"

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