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\centerline{\bg Chapter 1.  Mathematics and creativity}
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\noindent {\bf Grothendieck and generality}
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When asked to describe the nature of Grothendieck's mathematical contribution
-- not the results, but the style, the approach, the twist that made his
work uniquely recognizable -- the first thing mathematicians usually 
mention is the absolute generality visible in his approach to any
question.  Confronted with a specific problem, Grothendieck would 
tend to perceive it as a special case of some vast, 
far-reaching situation, and the cornerstone of his mathematical philosophy was 
that a general description of the vast situation would, once completed, 
naturally and effortlessly yield the properties of the situation, which would
then trickle down to give a solution to the particular problem.
Whereas mathematicians intent on proving a given result will often add to the
initial situation the extra hypotheses which, while restricting the scope,
will allow them to use arguments not valid in the general case, Grothendieck's
approach led him inversely to prove results with the least possible hypotheses 
on the objects he considered. 

What mathematicians ordinarily refer to as generalization is not 
precise or well-defined, although most mathematicians would agree without
the slightest dispute when something has been generalized.  One way to
generalize is to consider a specific, more or less well-understood statement 
on some set of objects, and prove that in fact it, possibly in some
modified form, applies to a larger set of objects.  Another type of 
generalization is to take a result which, in order for the proof to succeed,
necessitated a certain number of hypotheses at the start, and prove that
the statement still holds when these hypotheses are removed.  Accomplishing 
something of this kind is perceived as adding something to the original,
less general starting point, and the proof often appears more difficult in the
general situation, which is wider and broader.  The perceived difficulty of 
generalization lies partly in the unfamiliarity or abstraction of the 
generalized objects compared to the specific ones of the starting point, 
which being more classical or more familiar, provide a more thorough and 
accurate intuition and possibly a greater range of techniques.

In later chapters, we will discuss Grothendieck's generalizing approach to 
such topics as the Riemann-Roch theorem, the development of schemes, topos theory, 
etc.  That Grothendieck approached almost any new mathematical topic with the 
attitude that it was too specific and must be generalized is famous, one of 
the trademarks of his style.  He developed this tendency towards 
maximal generalization of every situation extremely early in his mathematical 
career, and it shows up in virtually every aspect of his research.  A
very colorful example is given in the following anecdote from the pen (and the
memory) of Prof. Jaap Murre of Leiden, who worked with Grothendieck in
the 1950s and 60s.
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{\it
    It happened precisely on the 18th of April, 1959.
  
Nico Kuiper had invited Grothendieck to give a lecture in Wageningen 
in the Netherlands. Nico Kuiper was then professor at the Agriculture 
University in Wageningen (we do have such a university in Holland); later 
Nico became professor at the University of Amsterdam and after that he became
-- as you undoubtedly know -- director of the IHES). He was a friend of 
Grothendieck; I think they knew each other from the Arbeitstagung in Bonn.

    There were (except Nico) almost no mathematicians in Wageningen, but
people had come from other places because Grothendieck was already 
well-known (Edinburgh 1958 for instance!).

     After the lecture in the morning (on cohomology of sheaves) and after
lunch, we went to Nico's house. I think I was, except for Grothendieck,
the only algebraic geometer present there (unbelievable now!),  so I did 
have ample opportunity to discuss with him.

      At that time there was a very important unsolved question `in the 
school of Weil' (to which I belonged), namely the behavior of the Picard
{\bf variety} if the original variety $X_t$, say, was varying in a system $T$.
In the beginning of the 50s Matsusaka, Weil himself and Chow had
constructed algebraically the Picard variety, but how did it vary when
the variety moved in a system and moreover -- and worse -- in characteristic 
$p> 0$? Igusa had discovered all kinds of ``pathological behavior", around 
1955. This was something which was not understood at all, neither from
the point of view of the school of Weil, nor from the somewhat different
point of view of Chevalley (and his school: Seshadri) who worked on this
Picard theory a little later (see for instance S\'em. Chevalley 1958/59).

So, during that discussion I asked Grothendieck whether he could
explain this behavior of the Picard variety.

He said that he had not yet studied that question carefully,
because this would be in chapter 12 of EGA. But he said that he would 
certainly settle that question and explain this behavior. He said
that his feeling was that ``those people" made too strict assumptions
and tried to prove too little. He intended to make less assumptions and
to prove a more precise theorem.

I said something like: ``Oh!, that sounds interesting'' (but I was very
sceptical!).

However then, in 1962, Grothendieck completely {\bf solved the question}
(not in chapter 12, of course, but in his two Bourbaki seminars).

With his theory of the Picard functor and the Picard {\bf scheme} he
proved a stronger theorem and explained precisely what was going on;
moreover things were not pathological at all from his point of view but
natural.

    I was at the IHES at that time (by the way the IHES was still in Paris 
then); I attended his Bourbaki lectures, and needless to say, 
I was very impressed !

     Of course, what was going on was that in characteristic $p > 0$, 
even if one started with nice varieties (i.e., reduced schemes), it could 
happen that the Picard functor was not representable by a variety but only by 
a scheme.

Undoubtedly, people did see in the mid 50's that one could generalize a 
lot of things to schemes, but Grothendieck saw that such a generalization was 
not only possible and natural, but {\bf necessary} to explain what was going on, 
even if one started with varieties.

}
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This episode illustrates Grothendieck's approach in a nutshell -- his 
seemingly more complicated and more general constructions were actually
intrinsically necessary to comprehend situations, and thus in fact simplified
them\footnote{${}^{*}$}{\sm Another example: ``L'an dernier encore, j'ai vu
reprocher \`a Contou-Carr\`ere de ne pas s'\^etre born\'e dans sa th\`ese
\`a se placer sur un corps de base au lieu d'un sch\'ema g\'en\'eral -- 
tout en lui conc\'edant quand m\^eme la circonstance att\'enuante que c'\'etait
surement sur les instances de son patron de circonstance qu'il avait d\^u
s'y r\'esoudre.   Celui qui s'exprimait ainsi \'etait pourtant suffisamment
dans le coup pour savoir que m\^eme en se bornant au corps des complexes,
les n\'ecessit\'es de la d\'emonstration forcent la main pour introduire
des sch\'emas de base g\'en\'eraux...'' (RS 279)}.
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Taken altogether, Grothendieck's body of work is perceived as an immense {\it tour de force}, 
an accomplishment of gigantic scope, and also extremely difficult both as
research and for the reader, due to the effort necessary to come to a 
familiar understanding of the highly abstract objects or points of view that 
he systematically adopts as generalizations of the classical ones.  All 
agree that the thousands of pages of his writings and those of his school,
and the dozens and hundreds of new results and new proofs of old results 
stand as a testimony to the formidable nature of the task. 

In this initial essay, however, without looking in detail at actual 
mathematics, we want to make a very important point: namely, that
the idea, the concept, or even the act of generalization can be viewed in
very different ways, and Grothendieck's own view of what he was doing, 
described in his own words, is totally different from the impression 
depicted above.  Grothendieck certainly did not feel that he was attempting
to use powerful techniques in order to obtain stronger results by generalizing.
What he perceived himself as doing was {\it simplifying} situations and
objects, by extracting the fundamental essence of their structure.  What
makes this seem so difficult for many people is a certain feeling of blindness 
in front of the new definitions, which by their abstraction and unfamiliarity 
deprive us of intuition\footnote{${}^{**}$}{\sm For instance, Grothendieck gives
this description of his efforts to reach a concrete understanding of an
interesting situation illustrated by one example: ``Je pense
notamment, dans le contexte justement de la cohomologie des vari\'et\'es
alg\'ebriques, \`a la d\'ecouverte par Griffiths de la fausset\'e d'une
id\'ee s\'eduisante qu'on avait eu longtemps sur les cycles alg\'ebriques,
\`a savoir qu'un cycle homologiquement \'equivalent \`a z\'ero avait un multiple
qui \'etait alg\'ebriquement \'equivalent \`a z\'ero.  Cette d\'ecouverte
d'un ph\'enom\`ene tout nouveau m'avait alors assez frapp\'e pour que je
passe bien une semaine de travail pour essayer de bien saisir l'exemple
de Griffiths, en transposant sa construction (qui \'etait transcendante,
sur le corps ${\bf C}$) en une construction ``aussi g\'en\'erale que
possible'', et valable notamment sur des corps de caract\'eristique
quelconque.  L'extension n'\'etait pas tout \`a fait \'evidente, \`a coups
(si je me rappelle bien) de suite spectrales de Leray et de th\'eor\`eme
de Lefschetz.'' (RS 289)}.

Apparently Grothendieck had no need for internal mental recourse to a concrete 
picture or a well-thumbed example.  His famous response `You mean like 57?' 
to somebody's suggestion `Pick a prime number' is often taken to illustrate
this. Michel Demazure recounts how with irrepressible optimism Grothendieck
would summarize his very first perception of a new situation with a conjecture,
-- which Serre would frequently observe was false for the most elementary 
example -- and J. Giraud [G] also describes the contrast between
Serre's precautionary style and Grothendieck's habit of tossing
out ideas with a casual ``\c Ca doit \^etre vrai!''

Grothendieck's independence from the concrete mathematical object was
well-known; as Jean-Pierre Serre, a lover of small and charming special cases, 
observed, it was not until his interest was sparked by dessins d'enfants 
that he finally understood the value and usefulness of actual examples.  
It would seem clear that this mental faculty of his to think and
reason independently of the {\it object} is the sign of a much more general
mental state studied by certain psychiatrists (such as J. Lacan), and not
unrelated to his attitude towards his own body and physical comfort.
Grothendieck's independence from the concrete objects of daily life was as
well-known to his contemporaries as his mathematical style; his diet of 
milk, cheese and bananas, his rudimentary house and clothing,
his sleeping on boards instead of a bed, his insistence on being operated 
without anesthesia: all these idiosyncrasies have become part of his legend.
Yet the two different forms of independence from the concrete have not
really been perceived as signs of one and the same mental state.

Because he saw the special features of specific mathematical objects and
situations as needlessly complicating the issue,
and general situations as being much simpler, he repeatedly described his
approach to mathematics as `childish' (`enfantine'), even `silly' or `babyish'
(`b\'eb\^ete'), and assumed that other people
did not adopt it because they would not stoop to something so absurdly simple.
These feelings about his own approach are expressed in his writings in
many different places.  For example, on his work on schemes, he writes:
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{\it L'id\'ee m\^eme de sch\'ema est d'une simplicit\'e enfantine -- si 
simple, si humble, que personne avant moi n'avait song\'e \`a se pencher si 
bas. Si ``b\'eb\^ete'' m\^eme, pour tout dire, que pendant des ann\'ees 
encore et en d\'epit de l'\'evidence, pour beaucoup de mes savants coll\`egues,
\c ca faisait vraiment ``pas s\'erieux"! Il m'a fallu d'ailleurs des mois 
de travail serr\'e et solitaire, pour me convaincre dans mon coin que ``\c ca 
marchait" bel et bien -- que le nouveau langage, tellement b\'eb\^ete, 
que j'avais l'incorrigible na\"\i vet\'e de m'obstiner \`a vouloir tester,
\'etait bel et bien ad\'equat pour saisir, dans une lumi\`ere et avec une 
finesse nouvelles, et dans un cadre commun d\'esormais, certaines des toutes 
premi\`eres intuitions g\'eom\'etriques attach\'ees aux pr\'ec\'edentes 
``g\'eom\'etries de caract\'eristique $p$".  C'\'etait le genre d'exercice, 
jug\'e d'avance idiot et sans espoir par toute personne ``bien inform\'ee", 
que j'\'etais le seul sans doute, parmi tous mes coll\`egues et amis, 
\`a pouvoir avoir jamais id\'ee de me mettre en t\^ete...} (RS 51)\footnote{${}^*$}{\sm The initials RS refer to R\'ecoltes et Semailles, and all page numbers 
correspond to the typed version available on the Grothendieck circle website.}

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And to explain the introduction of topos theory, his new conception of
topological space, he writes:
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{\it Voici donc l'id\'ee nouvelle. Son apparition peut \^etre vue comme une 
cons\'equence de cette observation, quasiment enfantine \`a vrai dire, 
que ce qui compte vraiment dans un espace topologique, ce ne sont nullement
ses ``points" ou ses sous-ensembles de points, et les relations de 
proximit\'e etc. entre ceux-ci, mais que ce sont les faisceaux sur cet espace, 
et la cat\'egorie qu'ils forment. Je n'ai fait, en somme, que mener vers sa
cons\'equence ultime l'id\'ee initiale de Leray -- et ceci fait, franchir 
le pas.

Comme l'id\'ee m\^eme des faisceaux (due \`a Leray), ou celle des 
sch\'emas, 
comme toute ``grande id\'ee" qui vient bousculer une vision inv\'et\'er\'ee 
des choses, celle des topos a de quoi d\'econcerter par son caract\`ere de
naturel, d'``\'evidence", par sa simplicit\'e (\`a la limite, dirait-on, 
du na\"\i f ou du simpliste, voire du ``b\'eb\^ete" -- par cette qualit\'e 
particuli\`ere qui nous fait nous \'ecrier si souvent: ``Oh, ce n'est 
que \c ca!", d'un ton mi-d\'e\c cu, mi-envieux; avec en plus, peut-\^etre, 
ce sous entendu du ``farfelu", du ``pas s\'erieux", qu'on r\'eserve souvent
\`a tout ce qui d\'eroute par un exc\`es de simplicit\'e impr\'evue. 
A ce qui vient nous rappeler, peut-\^etre, les jours depuis longtemps 
enfouis et reni\'es de notre enfance...} (RS 56)
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Given the usual impression one has of Grothendieck as a mathematician
at the summit of his profession, surrounded by admiring colleagues and
students, one may wonder who were those `well-informed people' and 
`knowledgeable colleagues' who `judged' his efforts to be `idiotic' and 
`hopeless'.  The name of Andr\'e Weil springs to mind.  Weil was 
undoubtedly somewhat allergic to Grothendieck's style of doing mathematics:
`...je me r\'ejouis fort de voir cette ``hypercohomologie'' donner des
r\'esultats tangibles en g\'eom\'etrie alg\'ebrique -- Weil sera furieux!'
Serre wrote to Grothendieck in February 1956, and `...cette difficult\'e 
que j'avais \`a m'ins\'erer dans le travail commun [\`a Bourbaki], ou les 
r\'eserves que j'ai pu susciter pour d'autres raisons encore \`a Cartan et \`a
d'autres, ne m'ont \`a aucun moment attir\'e sarcasme et rebuffade, ou
seulement une ombre de condescendance, \`a part tout au plus une fois ou
deux chez Weil (d\'ecid\'ement un cas \`a part)' (RS 141).  
But the feeling that in generalizing,
he was simplifying, and that this was not understood by almost any of his
colleagues, even the most willing, goes deeper than the pinch of resentment 
which transpires here, and must have contributed to the touch of
complacency expressed in these remarks (something felt
strongly by Jean-Pierre Serre, and denied with absolute sincerity by a 
surprised Grothendieck, cf. their correspondence). 

Beyond this, there is also present the echoes of a circle of deeper ideas 
about doing mathematics, and about creativity in general, which he took up in 
other places and explored in detail.  Indeed, it is natural to ask why 
Grothendieck was so easily able to think in these general terms, and what
it was about his approach that differed from his colleagues and made things
that were obvious to him seem difficult to them?  Because this biography
is an attempt to create a portrait of a mathematician, combining into a
coherent whole the psychological traits of the individual and his
astounding professional accomplishments, this question is a key one, and
its answer necessarily possesses manifold aspects, some of which we hope to 
illustrate both from Grothendieck's point of view and from that of an
outside observer.
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\noindent {\bf Grothendieck and the rising sea}
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In one of the most frequently cited passages from R\'ecoltes et Semailles,
Grothendieck compares his approach to solving a mathematical problems as
one of letting the obstacle become slowly absorbed, soaked and weakened
by the imperceptibly rising sea.
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{\it La mer s'avance insensiblement et sans bruit, rien ne semble se casser
rien ne bouge l'eau est si loin on l'entend \`a peine...Pourtant elle finit
par entourer la substance r\'etive, celle-ci peu \`a peu devient une
presqu'\^\i le, puis une \^\i le, puis un \^\i lot, qui finit par \^etre
submerg\'e \`a son tour, comme s'il s'\'etait finalement dissous dans
l'oc\'ean s'\'etendant \`a perte de vue...[...]

C'est `l'approche de la mer', par submersion, absorption, dissolution
-- celle o\`u, quand on n'est tr\`es attentif, rien ne semble se passer
\`a aucun moment: chaque chose \`a chaque moment est si \'evidente, et surtout,
si naturelle, qu'on se ferait presque scrupule souvent de la noter noir sur
blanc, de peur d'avoir l'air de bombiner, au lieu de taper sur un burin
comme tout le monde...C'est pourtant l\`a l'approche que je pratique d'instinct
depuis mon jeune \^age, sans avoir vraiment eu \`a l'apprendre jamais.}
(RS 502)
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Qualifying this approach as `feminine', or `yin', identifying as the
passive `acts' (or perhaps states) of waiting, watching, observing, 
letting ripen, gestating, he clearly contrasts this approach with another, 
perhaps more widespread perception of mathematics as a quintessentially active 
(and thus masculine): activity, consisting of  `solving problems', 
`penetrating the unknown', `unveiling secrets', acts which 
necessarily carry within them a seed of violence.
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{\it Prenons par exemple la t\^ache de d\'emontrer un th\'eor\`eme
qui reste hypoth\'etique (\`a quoi, pour certains, semblerait se r\'eduire
le travail math\'ematique).  Je vois deux approches extr\^emes pour s'y
prendre.  L'une est celle du {\bf marteau et du burin}, quand le probl\`eme
pos\'e est vu comme une grosse noix, dure et lisse, dont il s'agit 
d'atteindre l'int\'erieur, la chair nourrici\`ere prot\'eg\'ee par la
coque.  Le principe est simple: on pose le tranchant du burin contre
la coque, et on tape fort.  Au besoin, on recommence en plusieurs endroits
diff\'erents, jusqu'\`a ce que la coque se casse -- et on est content[...]

Je pourrais illustrer la deuxi\`eme approche, en gardant l'image de la
noix qu'il s'agit d'ouvrir. La premi\`ere parabole qui m'est venue \`a
l'esprit tant\^ot, c'est qu'on plonge la noix dans un liquide
\'emollient, de l'eau simplement pourquoi pas, de temps en temps on
frotte pour qu'elle p\'en\`etre mieux, pour le reste on laisse faire
le temps.  La coque s'assouplit au fil des semaines et des mois --
quand le temps est m\^ur, une pression de la main suffit, la coque
s'ouvre comme celle d'un avocat m\^ur \`a point!} (RS 501)
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These descriptions of his own work date from the 1980's, and thus come much 
later than the work itself, but they are nonetheless very accurate.
There was a consistent slowness, a refusal to hurry or to seek directly
for specific results in his approach.  The anecdote by Murre recounted
above illustrates this perfectly; results will ripen on the tree that 
he is patiently gardening, and fall off at the right time of their own 
juicy weight. 

This attitude, this incredible confidence that his rising sea would eventually
englobe and yield (easy, natural) proofs of all the open questions in the
domain, was perceived by many as one of those superhuman traits that
distinguished him from the common run of mathematicians; an incredible
confidence and ability to correctly predict his future ability to prove
something.  Jean-Pierre Serre scoffs at such an idea: `He wasn't predicting 
anything at all, he knew how to prove those things already!'  Steve Kleiman
points out, however, that Grothendieck announced the future purpose of
etale cohomology in Edinburgh in 1958, at a time where he certainly did not
yet know how to prove all the necessary results.  Either way, Grothendieck 
never intended to write up a proof until its proper place in the grand written 
work covering the entire theory had been reached, meaning that many of his 
theorems had to wait years before appearing in print, or else be written up 
and published by others, such as when Borel and Serre wrote up his proof of 
the Grothendieck-Riemann-Roch theorem.

One might summarize the quotations above by the remark that Grothendieck's 
approach to the mathematical object was not `figure out what's going on'
but `observe it deeply until its true nature is slowly revealed' -- an
activity in which {\it naming} the newly emerging objects, rather than
exploring their properties, played a fundamental role. 
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{\it Dans une telle situation, quand les choses elles-m\^emes nous soufflent
quelle est leur nature cach\'ee et par quels moyens nous pouvons le plus
d\'elicatement et le plus fid\`element l'exprimer, alors que pourtant beaucoup
de faits essentiels semblent hors de la port\'ee imm\'ediate d'une 
d\'emonstration, le simple instinct nous dit d'\'ecrire simlement noir sur
blanc ce que les choses nous soufflent avec insistance, et d'autant plus
clairement que nous prenons la peine d'\'ecrire sous leur dict\'ee!  Point
n'est besoin de se soucier de d\'emonstrations ou de constructions compl\`etes
-- s'encombrer de telles exigences \`a ce stade-l\`a du travail reviendrait
\`a s'interdire l'acc\`es de l'\'etape la plus d\'elicate, la plus essentielle
d'un travail de d\'ecouverte de vaste envergure -- celle de la naissance d'une
vision, prenant forme et substance hors d'un apparent n\'eant.  Le simple fait
{\bf d'\'ecrire}, de {\bf nommer}, de {\bf d\'ecrire} -- ne serait-ce 
d'abord que d\'ecrire des intuitions \'elusives ou de simples ``soup\c cons''
r\'eticents \`a prendre forme -- a un {\bf pouvoir cr\'eateur}.  C'est
l\`a l'instrument entre tous de la passion de conna\^\i tre, quand celle-ci
s'investit en des choses que l'intellect peut appr\'ehender.  Dans la d\'emarche
de la d\'ecouverte en ces choses-l\`a, ce travail en est l'\'etape cr\'eatrice
entre toutes, qui toujours pr\'ec\`ede la d\'emonstration et nous en donne
les moyens -- ou pour mieux dire, sans laquelle la question de ``d\'emontrer''
quelque chose ne se pose m\^eme pas, avant que rien encore de ce qui touche
l'essentiel n'aurait \'et\'e formul\'e et vu.  Par la seule vertu d'un
effort de formulation, ce qui \'etait informe prend forme, se pr\^ete \`a
examen, faisant se d\'ecanter ce qui est visiblement faux de ce
qui est possible, et de cela surtout qui s'accorde si parfaitement avec 
l'ensemble des choses connues, ou devin\'ees, qu'il devient \`a son tour
un \'el\'ement tangible et fiable de la vision en train de na\^\i tre.
Celle-ci s'enrichit et se pr\'ecise au fil du travail de formulation.  Dix 
choses soup\c conn\'ees seulement, dont aucune n'entra\^\i ne conviction,
mais qui mutuellement s'\'eclairent et se compl\`etent et semblent concourir
\`a une m\^eme harmonie encore myst\'erieuse, acqui\`erent dans cette harmonie
force de vision.  Alors m\^eme que toutes les dix finiraient par se r\'ev\'eler
fausses, le travail qui a abouti \`a cette vision provisoire n'a pas \'et\'e
fait en vain, et l'harmonie qu'il nous a fait entrevoir et qu'il nous a
permis de p\'en\'etrer tant soit peu n'est pas une illusion, mais une 
r\'ealit\'e, nous appelant \`a la conna\^\i tre.  Par ce travail, seulement,
nous avons pu entrer en contact intime avec cette r\'ealit\'e, cette harmonie
cach\'ee et parfaite. } (RS 286) 
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This attachment of his to the `true nature' of things was another typical 
feature of his approach.  Demazure likens his attitude towards the
contemplation of mathematical results or objects or examples to that
of a geologist picking up rocks at random and informing the ignorant
and astounded bystander that whereas this one means nothing at all, that
other one indicates the existence of some profound phenomenon a million
years ago.  `When Grothendieck looked at examples, he was searching for
hints about the true reality of things, ``nature's plan'' -- or ``God's 
plan''.  As for motives, it was the idea of perceiving in something that 
everyone else thinks is perfectly anodine, the sign of a much deeper,
hidden structure.'

If the `true nature' of things was a notion that had a clear meaning
for Grothendieck, it left some other mathematicians -- those
with a more democratic attitude towards objects, proofs and results -- cold.
Jean-Pierre Serre, for example, is ready to admire a difficult proof containing
unexpected twists and tricks, whereas such proofs made Grothendieck profoundly
uncomfortable, as containing something unnatural, forced, or incomplete.  
This difference in style was clearly noticeable already in 1961, when
over an argument about the inclusion of valuations into a Bourbaki volume,
Serre wrote to Grothendieck: `Je suis beaucoup moins ``entier'' que toi sur
ces questions (je n'ai pas de pr\'etention \`a conna\^\i tre ``l'essence''
des choses)...' a sentence which does denote a certain impatience with 
Grothen\-dieck's attitude.  

Pierre Cartier observed that when Grothendieck took interest in some 
mathematical domain that he had not considered up till then, finding a whole
collection of theorems, results and concepts already developed by others, 
he would continue building on this work `by turning it upside down'.  
Michel Demazure described his approach as `turning the problem into its own 
solution'.  In fact, Grothendieck's spontaneous reaction to whatever appeared 
to be causing a difficulty -- nilpotent elements when taking spectra or rings, 
curve automorphisms for construction of moduli spaces -- was to adopt and
embrace the very phenomenon that was problematic, weaving it in as an integral
feature of the structure he was studying, and thus transforming it from a
difficulty into a clarifying feature of the situation.  

All of the aspects of Grothendieck's mathematical approach discussed here:
the slow, broad approach, the search for the essence, the embrace without
reticence of a problem as its own solution -- all will be illustrated by
numerous cases and examples from his research in the coming chapters.

One might ask what it is that makes it so hard, or so rare, for other 
mathematicians to react this way.  Grothendieck suggests that the answer is 
undoubtedly a form of fear; fear that the unfamiliar will not bend docilely 
to the mathematician's will, fear that confronting the unknown will lead to 
unfortunate mathematical accidents such as error or total lack of progress,
fear of not obtaining recognizable results.  If there is one feature of
Grothendieck's personality to which he attributes his ability to have explored
and constructed so much that no one else had done before him, it is a total
lack of this kind of fear.
\vskip .8cm
\noindent {\bf Grothendieck and fearlessness}
\vskip .3cm
There is no doubt that ordinary mortals are psychologically equipped with 
a complex combination of more or less unconscious or unrealized fears and 
inhibitions.  These play a protective role in human development (as a
fear of heights will protect someone from venturing to dangerous places)
and most certainly, they play a role in the development of the individual
into a member of society with a sense of social behavior and a realization
that other people actually exist.

For reasons which may be partly inborn but are certainly partly due to family 
influence, it seems that Grothendieck was to all intents and purposes deprived 
of these fears and inhibitions, a phenomenon which accounts both for his
extreme social inadaptation and his absolutely free and uncomplexed approach
to mathematical research.

By all accounts, Grothendieck was pleasant, well-mannered, never unkind, 
generous, and already at twenty at ease with everyone, devoid of any sense
of awe or complex.  He spoke to the most prestigious mathematicians as equals
(Roger Godement says that had he encountered Hilbert personally, he would have 
cheerfully said `How are you?') and at the same time was particularly at 
ease with all people from the simplest
social class.  There are multiple anecdotes attesting this; 
his attachment to the domestic servant who lived with the Schwartz family, 
his kind attentiveness when Bill Messing's absence from a seminar reminded
him that the Messings were expecting a baby, his endless devotion to and 
generosity with his students, his suave, gentle speaking voice.  To be sure, 
when he did become angry, he was occasionally tempted to resort to his fists; 
Roger Godement recalls that
he claimed to be an excellent boxer, having had long practice with other
boys at school, and tells how he was once summoned to the local police station 
in Nancy after having roundly smacked a boy in the street who responded 
to a scolding by 
calling him `sale Boche'.  Serre and Raynaud distinctly recall seeing his 
fists clench and feeling a real moment of tension one day at the Institut 
Henri Poincar\'e, when Serre teased him about some mathematical assertion or 
other, and David Ruelle recalls that once, as he was driving Grothendieck down 
a rainy road away from the IHES, they were honked at and insulted by the 
driver just behind them; Grothendieck leaped out of the car and in a brief 
moment of panic Ruelle thought that he was about to witness a massacre
(fortunately he was able to calm the situation down).  Grothendieck's
girlfriend of a later period, Justine, mentions his actually attacking police 
officers, and even yielding to violent impulses with her, which 
actually led to the couple's early separation.  

Grothendieck's profound social inadaptation, however, is visible from 
other and subtler signs than these.
One of the most telling is that he felt that he had a biological rhythm
that was different from the ordinary 24 hour cycle, and this led him, 
throughout his life, to sleep, eat and work at strange, variable hours,
unrelated to what the rest of the world was doing.  That he should pursue his
own schedule in his own home would be normal enough, but in fact he attempted
to compel his wife and children to follow it with him, and as for inviting
him for a meal, Claudine Schwartz recalls that her parents Laurent and
Marie-H\'el\`ene had to calculate and negotiate at length to find one day
and one hour when Grothendieck's eating schedule would coincide with theirs.
Laurent Schwartz, according to Claudine, was quite fascinated with the
biological phenomenon, but at the same time, he sermonized Grothendieck, and
tried unsuccessfully to persuade him to adapt himself to family life and
to explain to him that society existed and that he was cutting himself
off from it.

When asked what Grothendieck was like as a young student in Nancy,
when he visited the Godements home regularly, Roger Godement states simply,
`Il \'etait un sauvage.'  According to Jean-Pierre Serre, `J'ai toujours 
senti qu'il \'etait comme une centrale nucl\'eaire, si on disait quelque chose
qui ne lui allait pas, il pouvait exploser, et ce serait grave, il valait mieux \'eviter.'
The impression of those who knew him best was that in spite of a pleasant 
exterior and what soon became a highly respected professional position,
there was something deeply abnormal and disquieting about him, in his total
lack of real comprehension of other people (quote from a 1995 letter: `Je ne 
m'attendais pas \`a recevoir encore de signe de vous, et me r\'ejouis de 
m'\^etre encore tromp\'e, comme c'est d'ailleurs pratiquement toujours le cas 
quand je me hasarde \`a quelque pr\'evision concernant un de mes semblables')
and in his excessive devotion to his mathematical studies, which 
occupied him for some sixteen hours every day, which prevented any of the 
people who surrounded him from approaching him close enough to form an 
intimate and lasting friendship.  It is telling that although in
R\'ecoltes et Semailles and his other writings, Grothendieck very frequently
refers to his friends, few or none of those people actually now describe 
themselves as having truly been friends with him, in the deep sense of the
word friendship.  Somewhere, Grothendieck must have been aware of this,
since sadly but not surprisingly, he writes as a description of
his entire life until the moment in 1976 in which he suddenly discovered 
meditation: `Je me sentais spirituellement absolument seul de mon esp\`ece, et 
n'arrivais \`a me reconna\^\i tre dans aucun groupe humain, ni dans aucun
autre \^etre' (Clef 104).

If Grothendieck did show kindness and good manners, if he did say `bonjour'
and `merci', it is very probably because of the five years he spent living 
with the Heydorn family.  Certainly the bringing up he received from his parents
before the age of five, and the period he spent living with his mother 
from 1939 to 1948 were powerful influences in driving him
away from what his parents considered the bourgeois aspects of society.
Transgression of traditional social rules and boundaries,
viewed by Grothendieck's parents (and in general, by all who shared
these early anarchist ideals which developed into the movement associated
with `the 60's' in the United States and `May 1968' in France)
as the noblest rebellion in the history of mankind,
may have been the most important and visible legacy that Sascha 
Tanaroff\footnote{${}^*$}{\sm as Grothendieck's father was known 
and identified on his papers; Grothendieck, however, asserts that this was
a false name and that his father's name at birth was
Shapiro} and Hanka Grothendieck left their son.

Grothendieck's father Sascha was a political anarchist, a true one.  His mother was 
less political, but her temperament was violent, tempestuous and difficult, 
and by nature more anarchist than any theoretical anarchist.  No political 
or social stream
could contain her.  She adopted the anarchist hatred of authority and
capitalism, but had no real interest in their wider goals of creating a
society of mutual support and friendship.  In fact, she was perhaps more of a
nihilist than an anarchist; rejection of bourgeois values of respectability
and comfort, rejection of social values of politeness and respect, rejection 
of any consciously imposed values at all; profound admiration of passion, love,
freedom, strength, will-power, genius.  During her lifetime, she purposely
violated every boundary she encountered; broke every spoken and unspoken
rule with a kind of vicious
pleasure, destroying her own life and the lives of everyone around her 
in the process.  The picture of her life which she somehow imposed on both her
children for many decades, was that of passionate lover to her anarchist
hero Sascha and heroic mother to extraordinary children.  The reality was
a scene of ceaseless conflict and quarreling with her lover, the abandonment of 
her children, and finally, when reunited with them, the absolute and
shocking rejection of every aspect of motherhood.  Grothendieck described 
living with her during the last years of her life (she died in 1957) as hell.
And yet, there is no doubt that her influence and her powerful personality
left him something besides a decades-long deviant admiration for purely 
masculine values, which corresponded to her character rather than to his.
It could well be her transgressive influence that produced a 
child, then a young man, and an adult, so entirely free from fear and from any 
sense of limits, of boundaries, of `what others might think' and of
`when to stop'.  Grothendieck most probably owes the extreme aspects of his 
nature, and the total freedom from fear and complex, what
he describes as the complete absence of an inner division, to his parents.

In one of the most fascinating passages of R\'ecoltes et Semailles, 
Grothendieck recounts how he himself came to realize this feature
of his nature, and to begin to analyse its possible origin.
\vskip .3cm
{\it Il y a quatre ans, j'ai pour la
premi\`ere fois senti et mesur\'e la port\'ee d'une chose dans ma vie \`a
laquelle je n'avais jamais song\'e, qui toujours m'avait sembl\'e aller de
soi: c'est que mon identification \`a mon p\`ere, dans mon enfance, n'a 
{\bf pas} \'et\'e marqu\'ee par le conflit -- qu'en aucun moment de mon
enfance, {\bf je n'ai craint ni envi\'e mon p\`ere}, tout en lui vouant un
amour sans r\'eserve.  Cette relation-l\`a, la plus profonde peut-\^etre
qui ait marqu\'e ma vie (sans m\^eme que je m'en rende compte avant cette
m\'editation d'il y a quatre ans), qui dans mon enfance a \'et\'e comme la
relation \`a un autre moi-m\^eme \`a la fois fort et bienveillant -- cette
relation n'a pas \'et\'e marqu\'ee par le sceau de la division et du conflit.
Si, \`a travers toute ma vie bien souvent d\'echir\'ee, la connaissance de la
force qui repose en moi est rest\'ee vivante; et si, dans ma vie nullement
exempte de peur, je n'ai pas connu la peur ni d'une personne ni d'un
\'ev\'enement -- c'est \`a cette humble circonstance que je le dois, ignor\'ee
encore jusqu'au del\`a de mes cinquante ans.  Cette circonstance a \'et\'e
un privil\`ege sans prix, car c'est la connaissance intime de la force
cr\'eatrice en sa propre personne qui {\bf est} aussi cette force, qui lui
permet de s'exprimer librement selon sa nature, par la cr\'eation -- par une
vie cr\'eatrice.} (RS 376)
\vskip .3cm
This description is all the more remarkable in that Grothendieck's family
(his parents, his older sister Maidi, and himself) was by all accounts
torn by perpetual quarrelling and conflict throughout the five years 
during which they actually lived together, a state of affairs witnessed 
by Hanka's writings and Maidi's memories (as recounted by her daughter Diana)
as well as by Grothendieck himself more than once:
\vskip .3cm
{\it Les trois \^etres les plus proches, qui ensemble ont constitu\'e comme
la matrice de mes premi\`eres ann\'ees, \'etaient d\'echir\'es par le conflit,
opposant chacun d'eux et \`a lui-m\^eme, et aux deux autres: conflit
insidieux, au visage impassible entre ma m\`ere et ma s\oe ur, et conflit
aux violents \'eclats entre mon p\`ere et ma m\`ere d'un c\^ot\'e, ma
s\oe ur de l'autre, qui chacune pour son propre compte (et sans que personne
du vivant de mes parents ait jamais fait mine de s'en apercevoir...) le
faisait marcher \`a sa fa\c con.  La chose myst\'erieuse, extraordinaire,
c'est qu'entour\'e ainsi par le conflit en ces ann\'ees les plus sensibles,
les plus cruciales de la vie, celui-ci soit rest\'e {\bf ext\'erieur}
\`a moi, qu'il n'ait pas vraiment ``mordu'' sur mon \^etre en ces ann\'ees-l\`a
et ne s'y soit install\'e \`a demeure....L\`a, je reviens \`a cette
``chose myst\'erieuse'', l'{\bf absence de division en moi}, en ces premi\`eres
ann\'ees de ma vie.

Peut-\^etre le myst\`ere n'est plus pour moi en cette absence, mais plut\^ot
en ceci: que mes parents, mon p\`ere comme ma m\`ere, m'aient chacun alors
{\bf accept\'e dans ma totalit\'e}, et totalement: dans ce qui en moi
est ``viril'', est ``homme'', et dans ce qui est ``femme''.  Ou pour le
dire autrement: que mes parents, d\'echir\'es l'un et l'autre par le conflit,
reniant chacun une partie essentielle de leur \^etre -- incapable chacun
d'une ouverture aimante \`a lui-m\^eme et \`a l'autre, comme d'une ouverture
aimante \`a ma s\oe ur...que n\'eanmoins ils aient trouv\'e une
telle ouverture, une acceptation sans r\'eserve, vis \`a vis de moi leur fils.}
(RS 450)
\vskip .3cm
That Grothendieck's approach to life, to work, to society, to other people, and
to mathematics is absolutely whole and entire, monolithic, inaccessible to
influence or modification from the outside, is obvious.  Whether this 
indivision is really due to the loving acceptance of his parents during his
earliest childhood, or whether contrarily he managed to preserve deeply loving,
happy memories of his first five years in a conflict-ridden family, and to
describe himself as having been able to remain outside these conflicts,
thanks to some inborn native essential wholeness, is certainly not clear.
And memories and impressions are subjective and may also change over time.
However, the strong feeling he expresses of his own integral nature and lack
of fear is borne out by his behavior throughout his life, both negatively
and positively, with respect to his relationship with people and with
mathematics.

When discussing the issue of fear and inhibition in mathematics, Grothendieck 
perceives and describes with frightening clarity its destructive effects on 
creativity.  
\vskip .3cm
{\it Craindre l'erreur et craindre la v\'erit\'e est une seule et m\^eme
chose!  Celui qui craint de se tromper est impuissant \`a d\'ecouvrir.}  
(RS 129)
\vskip .3cm
For him, however, the fear of observation and discovery forms a vicious circle 
with the power of vanity which, building an admirable self-image, represses
the fear but simultaneously intensifies it, as the possibility of discovering
something unpleasantly different from the reassuring self-image becomes
increasingly dangerous.  This circle is one of the essential topics 
throughout his writings.
\vskip .3cm
{\it Mon propos dans R\'ecoltes et Semailles a \'et\'e de parler de l'un et
de l'autre aspect -- de la pulsion de connaissance, et de la peur et de ses
antidotes vaniteux.  Je crois ``comprendre'', ou du moins
{\bf conna\^\i tre} la pulsion et sa nature.  (Peut-\^etre un jour
d\'ecouvrirai-je, \'emerveill\'e, \`a quel point je me faisais illusion...)
Mais pour ce qui est de la peur et de la vanit\'e, et les insidieux blocages
de la cr\'eativit\'e qui en d\'erivent, je sais bien que je n'ai pas \'et\'e
au fond de cette grande \'enigme.} (RS 28)
\vskip .3cm
According to his analysis, the role of the ego is to avoid any appearance
of failure in the eyes of the individual himself, and the fear of failure
is above all the fear of tarnishing the self-image, and thereby losing control 
over the perception of the self.
\vskip .3cm
{\it Souvent une peur secr\`ete fait barrage au ``plaisir'' m\^eme
que l'on croit rechercher, effray\'e qu'on est par la pr\'esence toute
proche d'une force inconnue et redoutable, qui risque (si on n'y veille...)
de balayer comme f\'etu de paille Celui en nous qui \`a tout prix tient
\`a garder ``le contr\^ole''.  }  (RS 479)
\vskip .3cm
Thus, for Grothendieck, fear of the collapse of one's stable and reassuring
vision (whether of the self or of the outer world, or of a mathematical 
situation) is what paralyses creativity and prevents profound observation; 
it is a purely negative and absolutely destructive phenomenon, and its
absence -- a kind of confidence or self-assurance -- is the necessary 
precondition, and seemingly the only one, for true creative work.  
\vskip .3cm
{\it Cette {\bf assurance}-l\`a est l'une des faces d'une disposition 
int\'erieure,
dont l'autre face est une {\bf ouverture au doute}: une attitude de curiosit\'e
excluant toute crainte, vis \`a vis de ses propres erreurs, qui permet de les
d\'epister et de les corriger constamment.  La condition essentielle de cette
double assise, de cette {\bf foi} indispensable pour accueillir le doute comme
pour d\'ecouvrir, est l'absence de toute peur (qu'elle soit apparente ou
cach\'ee) au sujet de ce qui ``sortira'' de la recherche entreprise -- de toute
peur, notamment, que la r\'ealit\'e que nous nous appr\^etons \`a d\'ecouvrir
bouscule nos certitudes ou convictions, qu'elle ne d\'esenchante nos
espoirs.  Une telle peur agit comme une paralysie profonde de nos facult\'es
cr\'eatrices, de notre pouvoir de renouvellement.} (RS 430)
\vskip .3cm
There is a great deal of truth in this vision.  What is striking, 
though, is that Grothendieck seems to totally lack any understanding of the 
protective reasons underlying this psychological structure that he
calls `fear and vanity', which is so common and so widespread that humanity 
has known only a handful of exceptions (a list of some of these
`mutations' -- Darwin, Freud, Walt Whitman, Gandhi -- is given in the 
`Mutants' section of the Notes to La Clef des Songes).  While clearly 
perceiving the human need for a positive self-image, for reassuring and settled 
views, and for a certain stability in one's image of self and the world, he 
qualifies them contemptuously as nothing more than our `sempiternelle fringale de certitudes et de s\'ecurit\'es, l'instinct du troupeau \`a la recherche
du berger' (Clef 161) which constitutes a fundamental repressive barrier to 
creativity in every individual, the necessity condition for creativity being
an openness to complete, profound and continual renewal.

Similarly, Grothendieck was entirely unable, as he showed again and
again throughout the years of his ecological activities, to comprehend the 
capacity of ordinary people to remain optimistically indifferent to the
terrifying danger, perceived by some as imminent, of human destruction by 
nuclear weapons, not even to mention popular indifference to the actual
ills and evils of the world at the present moment: wars, genocide, famine,
starvation all taking place more or less under our eyes and forced on our
attention on a daily basis by newspapers and information, as we go about our
lives.  This attitude, this indifference, appear to him to be quite mad
(and once his point of view has been made clear and absorbed, a part of 
oneself does undoubtedly come to share it).  Grothendieck's eventual
reclusion was in part due to the barrier created between himself and the rest
of humanity by this difference, after many years of unsuccessfully attempting
to share his vision, first by political and ecological activity,
then by prophetic communications, none of which encountered any real wave
of sympathy or success; in a 1995 letter, he wrote `vous \^etes dans
la Maison des Fous, que vous imaginez encore (comme moi-m\^eme nagu\`ere)
\^etre ``le monde'', et me dites ce que ``les gens'' y pensent, et y font.
Je suis dehors, ``au grand large''...' 

The psychological safeguard mechanisms which place a veil of distance 
between ordinary people and the horrors that surround them, which allow them
to continue seeking their own comfort in their daily lives (even while paying 
lip service to the right feelings) are apparently totally lacking
in Grothendieck's mental makeup.  The humble consolations of the
ego: `the world is a dreadful place, but at least I have succeeded in avoiding
the worst of the suffering, in accomplishing something that will last, in 
providing a happy and healthy home for my children' -- these appear to
Grothendieck as no more than a disastrously destructive set of blinders
that we voluntarily wear out of the sheer fear of {\it looking at things the
way they are}, whether within ourselves or outside in the larger world. 
To him, the way things are, the natural way in which they occur -- even the
worst horrors that individual and collective humanity has perpetrated --
have their own necessary meaning, significance and harmony, and the essential 
creative act lies in taking the time to seek for this, to look, to watch, to 
observe, to feel and see, all the while patiently and unceasingly
peeling away from our overactive minds the illusions that spring from its
excessive need to interfere with observation by conscious thought.   And in
fact, a great deal of his later `prophetic' writings and recordings of
religious communications are concerned precisely with the cosmic explanation
for the -- otherwise absolutely intolerable -- existence of evil.
\vskip .8cm
\noindent {\bf The child-explorer and the child-builder} 
\vskip .4cm
In one of the most important passages in R\'ecoltes et Semailles, one which 
introduces the key image of Grothendieck's conception of creativity,
he introduces the theme of the `child'; the actual child, as the example
of creativity given free rein, and the image of the child within the adult, 
embodying its continuation throughout life.
\vskip .3cm
{\it La d\'ecouverte est le privil\`ege de l'enfant.  C'est du petit enfant
que je veux parler, l'enfant qui n'a pas peur encore de se tromper, d'avoir
l'air idiot, de ne pas faire s\'erieux, de ne pas faire comme tout le monde.
Il n'a pas peur non plus que les choses qu'il regarde aient le mauvais
go\^ut d'\^etre diff\'erentes de ce qu'il attend d'elles, de ce qu'elles
devraient \^etre, ou plut\^ot: de ce qu'il est bien entendu qu'elles
{\bf sont}.  Il ignore les consensus muets et sans failles qui font partie
de l'air que nous respirons -- celui de tous les gens sens\'es et bien connus
comme tels.  Dieu sait s'il y en a eu, des gens sens\'es et bien connus
comme tels, depuis la nuit des \^ages!

Nos esprits sont satur\'es d'un ``savoir'' h\'et\'eroclite, enchev\^etrement
de peurs et de paresses, de fringales et d'interdits; d'informations \`a
tout venant et d'explications pousse-bouton -- espace clos o\`u viennent
s'entasser informations, fringales et peurs sans que jamais ne s'y engouffre
le vent du large.  Exception faite d'un savoir-faire de routine, il semblerait
que le r\^ole principal de ce ``savoir'' est d'\'evacuer une perception
vivante, une prise de connaissance des choses de ce monde.  Son effet est
surtout celui d'une inertie immense, d'un poids souvent \'ecrasant.}
(RS 127)
\vskip .3cm
For Grothendieck, the ultimate creative being is the child, and this is 
not a metaphor.  Believing as he does that the essential creative 
force exists in every person, his view of creativity is not
to be confused with production of any actual outcome, but expresses an 
absolutely unfettered ability to observe and feel the flow of the natural 
world. The spontaneous response of the child to this flow is to
thrust his hands into it and begin exploring,
touching, feeling and forming.  Although Grothendieck does not explicitly
mention beaches, having perhaps never spent any time on one, any tiny child on 
a beach provides an ideal illustration of his image, the infinitely supple 
plasticity of the wet sand fitting perfectly with the varying imagination of 
the child, engaged in an activity of which digging, shaping,
destroying, wallowing, building, and simply grovelling are all merely different
aspects, of equal value to the child: 
`l'enfant-qui-aime-\`a-explorer-les-choses, \`a aller fouiner et s'enfouir
dans les sables ou dans les vases boueuses et sans nom, les endroits les
plus impossibles et les plus saugrenus...' (RS 63)

Perhaps the very crux of the matter is that creativity, for Grothendieck,
is identified with the accepting and comprehending observation of the 
mysterious ways of nature -- observation made with respect and love, 
absolutely devoid of judgment -- {\it this is creativity}, and it is
completely independent of whether anything material is actually produced.  
\vskip .3cm
%{\it Fundamental quote where he says this explicitly (Clef?)}
%\vskip .3cm
In a footnote full of charm, Grothendieck describes the enchantment of 
his tiny daughter as a quintessential moment of discovery:
\vskip .3cm
{\it Elle devait avoir un an ou deux, quelqu'un venait de
jeter des granul\'es dans un bocal de poissons rouges.  Les poissons 
s'empressaient \`a qui mieux mieux de nager vers eux, la gueule grande
ouverte, pour ingurgiter les minuscules miettes jaunes en suspension qui
descendaient lentement dans l'eau du bocal.  La petite ne s'\'etait jamais
rendue compte avant que les poissons mangeaient comme nous.  C'\'etait
en elle comme un \'eblouissement soudain, s'exprimant en un cri de pur
ravissement: ``Regarde maman, {\bf ils mangent}!''  Il y avait de quoi
s'\'emerveiller en effet -- elle venait de d\'ecouvrir en un \'eclair
subit un grand myst\`ere: celui de notre parent\'e \`a tous les autres \^etres
vivants...} (RS 200)
\vskip .3cm
With all this, however, the productive aspect of creativity is of course not absent from 
Grothendieck's view.  He perceives it as another facet: as a complement to 
`l'enfant-qui-explore', he introduces `l'enfant-qui-b\^atit'.  The first,
initially introduced as a masculine type (`le pionnier, l'explorateur')
is transformed, as he examines more closely the activity he is trying to 
describe, into a feminine personage (`une s\oe ur des mares, de la pluie, 
des bruines et de la nuit, silencieuse et quasiment invisible \`a force de
s'effacer dans l'ombre' (RS 64)), so that the two facets of creativity 
together form one of the yin-yang pairs in which both are necessary to achieve 
a harmonious balance.
\vskip .3cm
{\it Quand je construis, am\'enage, ou que je d\'eblaie, nettoie, ordonne,
c'est le ``mode'' ou le ``versant'' ``yang'', ou ``masculin'' du travail
qui donne le ton.  Quand j'explore \`a t\^atons l'insaisissable, l'informe,
ce qui est sans nom, je suis le versant ``yin'', ou ``f\'eminin'' de mon \^etre.

Il n'est pas question pour moi de vouloir minimiser ou renier l'un ou l'autre
versant de ma nature, essentiels l'un et l'autre -- le ``masculin'' qui 
construit et qui engendre, et le ``f\'eminin'' qui con\c coit, et qui abrite
les lentes et obscures gestations.  Je ``suis'' l'un et l'autre --
``yang'' et ``yin'', ``homme'' et ``femme''.  }  (RS 63)
\vskip .3cm
Yet Grothendieck admits that he himself devoted the greatest part of his 
energy during his established professional career
to the building rather than the exploring aspect of research.
\vskip .3cm
{\it Mon approche des math\'ematiques, depuis l'\^age de dix-sept ans quand j'ai
commenc\'e \`a m'y investir \`a fond, a \'et\'e de me poser des grandes
{\bf t\^aches}.  C'\'etaient toujours, d\`es le d\'ebut, des t\^aches 
de ``mise en ordre'', de grand nettoyage.  Je voyais un apparent chaos, une
confusion de choses h\'et\'eroclites ou de brumes parfois impond\'erables,
qui visiblement devaient avoir une essence commune et receler un ordre, une
harmonie encore cach\'ee qu'il s'agissait de d\'egager par un travail
patient, m\'eticuleux, souvent de longue haleine.  C'\'etait un travail
souvent \`a la serpill\`ere et au balai-brosse, pour la grosse besogne
qui d\'ej\`a absorbait une \'energie consid\'erable, avant d'en venir aux
finitions au plumeau, qui me passionnaient moins mais qui avaient aussi leur
charme et, en tous cas, une \'evidente utilit\'e[...]La fid\'elit\'e \`a
mes ``t\^aches'' m'interdisait d'ailleurs des \'echapp\'ees trop lointaines,
et je rongeais mon frein dans une impatience d'\^etre arriv\'e au bout
de toutes et m'\'elancer enfin dans l'inconnu, le vrai -- alors que la
dimension de ces t\^aches \'etait devenue telle d\'ej\`a, que pour les 
mener \`a bonne fin, m\^eme avec l'aide de bonnes volont\'es qui avaient
fini par arriver \`a la rescousse, le restant de mes jours n'y aurait
pas suffi!} (RS 201)
\vskip .3cm
In certain passages, the feeling of impatience he mentions here is expressed 
as something much stronger; his compulsion to terminate planned tasks instead
of exploring unknown territory is expressed as a heavy burden.
\vskip .3cm
{\it Souvent je rongeais mon frein d'\^etre retenu ainsi, comme par un poids
tenace et collant, avec ces interminables t\^aches qui (une fois vu l'essentiel)
s'apparentaient plus pour moi \`a de l'``intendance'', qu'\`a une lanc\'ee dans
l'inconnu.  Constamment je devais retenir cette pulsion de m'\'elancer de
l'avant -- celle du pionnier ou de l'explorateur, parti \`a la d\'ecouverte
et \`a l'exploration de mondes inconnus et sans nom, m'appelant sans cesse
pour que je les connaisse et les nomme.  Cette pulsion-l\`a, et l'\'energie
que j'y investissais (comme \`a la d\'erob\'ee, quasiment!), \'etaient
constamment \`a la portion congrue.} (RS 61)
\vskip .3cm
And in the famous last sentence from the {\it Esquisse d'un Programme}, he
calls himself literally a prisoner of this compulsion, and expresses his 
release from it as a blessed escape.
\vskip .3cm
{\it Aujourd'hui je ne suis plus, comme nagu\`ere, le prisonnier volontaire
de t\^aches interminables, qui si souvent m'avaient interdit de m'\'elancer
dans l'inconnu, math\'ematique ou non.  Le temps des {\bf t\^aches} pour moi
est r\'evolu.  Si l'\^age m'a apport\'e quelque chose, c'est d'\^etre plus
l\'eger.}
\vskip .3cm
Jean-Pierre Serre has pointed out that one of the dominating features of the 
impression Grothendieck made on others during the 15 or 20 years of his 
established professional career was of being impelled to do mathematics,
not by a sense of joy, delight, or beauty, but by a sense of duty.  
Grothendieck would not fully agree; if he did not always succeed in 
communicating his powerful sense of the search for beauty in mathematics,
he most certainly felt it deeply within himself and considered it as his
guide: `C'\'etait un sens aigu de la ``beaut\'e'', surement, qui \'etait mon
flair et ma seule boussole' (RS 202).  The sense of beauty, however,
is a very subjective thing; he describes his as `la recherche constante
d'une coh\'erence parfaite, d'une harmonie compl\`ete que je devinais 
derri\`ere la surface turbulente des choses', which is very different from
the kind of delight in mathematical objects or arguments which can be
perceived as pretty (`jolis'), unexpected or charming that 
Serre himself might evince.  Grothendieck's description of his own sense
of beauty in mathematics already provides the beginning of an explanation of
his compulsive devotion to completing gigantic panoramic tasks.
He himself made a conscious effort to understand why he followed this tendency
in himself to the point of restraining himself from the kind of 
free exploration which gave him as much joy or more.  
Attributing it to egotic forces -- his giving greater
value to the `masculine' values of completed published work than to the
`feminine' values of risky and uncertain exploration -- he contrasts 
it with the attitude of other mathematicians, who work for the sheer joy 
of the thing, happily lacking this binding sense of responsibility:
\vskip .3cm
{\it Ma relation \`a la math\'ematique (et surtout, \`a la production
math\'ematique) \'etait fortement investie par l'ego, et ce n'\'etait pas le
cas chez Mike [Artin].  Il donnait vraiment l'impression de faire des maths
comme un gosse qui s'amuse, et sans pour autant oublier le boire et le manger.
} (RS 161)
\vskip .3cm
The personal value system which encouraged `achievement' over `exploration'
that Grothendieck followed for the first part of his life also appeared to 
him to correspond to the values of his milieu:
\vskip .3cm
{\it Les consensus en vigueur m'encourageaient \`a investir le plus clair
de mon \'energie dans l'autre versant, dans celui qui s'incarne et s'affirme
dans des ``produits'' tangibles, pour ne pas dire finis et achev\'es -- des
produits aux contours bien tranch\'es, attestant de leur r\'ealit\'e avec
l'\'evidence de la pierre taill\'ee.
Je vois bien, avec le recul, comment ces consensus ont pes\'e sur moi, et aussi
comment j'ai ``accus\'e le poids'' -- en souplesse!  La partie ``conception''
ou ``exploration'' de mon travail \'etait maintenue \`a la portion congrue
jusqu'au moment encore de mon d\'epart.} (RS 63)
\vskip .3cm
There was, however, a generosity in his attitude even if he simultaneously
felt that
he had been impelled by forces of ego or vanity.  At the time of his work, he
very clearly felt that it was destined for the common good; an absolutely
necessary piece of work that simply must be done by anyone who could; 
by himself alone if no one else was available to help.  In a famous
letter to Henri Cartan, from October 1961, protesting the necessity for
graduate students in mathematics to leave for the mandatory two years'
military service, he wrote: `J'ai p\'eniblement d\'ecroch\'e pour mon
s\'eminaire de g\'eom\'etrie alg\'ebrique \`a l'IHES quatre ou cinq
ex-normaliens, qui commencent \`a avoir de vagues lueurs, et dont un
ou deux semblaient sur le point de d\'emarrer sur du travail utile,
voire urgent, savoir Verdier et Giraud.  Bernique, sauf erreur tous les
deux, et en tous cas Verdier, devancent l'appel, et si \c ca se trouve
quelqu'un d'autre fera le boulot \`a leur place (moi-m\^eme s'il le faut).'
He perceived his planned work as absolutely necessary; not his own personal
research, but foundational work essential to the further development of a 
subject in need of renewal.  In August 1959, still in the early flush of
starting the EGA's, he wrote a wildly optimistic plan to Serre:
\vskip .3cm
{\it J'esp\`ere arriver dans l'ann\'ee prochaine \`a une th\'eorie
satisfaisante du groupe fondamental, et achever la r\'edaction des chapitres
IV, V, VI, VII (ce dernier \'etant le groupe fondamental), en m\^eme
temps que des cat\'egories.  Dans deux ans r\'esidus, dualit\'e, 
intersections, Chern, Riemann-Roch.  Dans trois ans cohomologie de Weil,
et un peu d'homotopie si Dieu veut.  Et entre-temps, je ne sais quand,
le ``grand th\'eor\`eme d'existence'' avec Picard etc., un peu de
courbes alg\'ebriques, les sch\'emas ab\'eliens.  Sans difficult\'es
impr\'evues ou enlisement, le multiplodoque devrait \^etre fini d'ici
3 ans, ou 4 ans maximum.  On pourra commencer \`a faire de la
g\'eom\'etrie alg\'ebrique!}
\vskip .3cm 
As Demazure points out, the last sentence `On pourra commencer \`a faire
de la g\'eom\'etrie alg\'ebrique' must be read as it is meant: not that
{\it Grothendieck} (and his associates) would be able to start working on
algebraic geometry, but that the mathematical community as a whole would be
enabled to make serious advances in the subject, thanks to the foundational
work that Grothendieck and Dieudonn\'e were undertaking in the service
of all mathematicians, of mathematics itself.  The strong sense of duty and 
public service was felt by everyone around Grothendieck\footnote{${}^*$}{\sm As
 Steve Kleiman notes: ``I can attest to this statement 
from my own experience.  Grothendieck told me to read a little EGA every day 
in order to acquire a sense for its contents, because he and Dieudonn\'e 
were writing EGA as a service and we should appreciate this fact and take 
advantage of it.''}, and it is absolutely
not in contradiction with the simultaneous sense of personal involvement
and pride: 
\vskip .3cm
{\it La force principale, le ``drive'' qui \'etait derri\`ere 
l'investissement que je faisais en mes \'el\`eves en g\'en\'eral, dans la
premi\`ere p\'eriode des ann\'ees soixante, c'\'etait le d\'esir de trouver
``{\bf des bras}'' pour r\'ealiser des ``{\bf t\^aches}'' que mon instinct me
d\'esignait comme urgentes et importantes...Cette ``importance'' surement
n'\'etait pas purement subjective, ce n'\'etait pas une simple question
``de go\^uts et de couleurs''...Pourtant, pour ce qui est de ce ``drive'',
de cette force de motivation en moi qui me poussait vers la r\'ealisation
des t\^aches, ce n'\'etait pas une certain importance ``objective'' qui \'etait
en jeu -- alors que ``l'importance'' de la conjecture de Fermat, de
l'hypoth\`ese de Riemann ou de celle de Poincar\'e me laissaient parfaitement
froid, que je ne les ``sentais'' pas vraiment.  Ce qui distinguait ces
t\^aches de toutes autres, dans ma relation \`a elles, c'est que c'\'etaient
{\bf mes} t\^aches; celles que j'avais senties, et faites miennes...C'est le
lien profond entre celui qui a con\c cu une chose, et cette chose...Il me
para\^\i t profond\'ement enracin\'e dans la nature du ``moi'', et de
nature universelle.}  (RS 325)
\vskip .3cm
If, in the yin-yang aspects of creativity, Grothendieck chose for years
of his life
to privilege the yang, he does not doubt that it was at the deepest level a
question of personal glory and self-aggrandizement: `L'investissement
dans mes t\^aches \'etait de nature \'egotique...Surement la r\'ealisation
de ces t\^aches \'etait surtout, pour le ``moi'', un moyen de s'agrandir,
par la r\'ealisation d'une \oe uvre d'ensemble aux vastes proportions...
A partir d'un certain moment dans ma vie de math\'ematicien, il y a eu
cette ambigu\"\i t\'e constante d'une cohabitation, d'une interp\'en\'etration
\'etroite entre ``{\bf l'enfant}'' et sa soif de conna\^\i tre et de 
d\'ecouvrir, son \'emerveillement en les choses entrevues et en celles
examin\'ees de pr\`es, et d'autre part le {\bf moi}, le ``{\bf patron}'',
se r\'ejouissant de ses \oe uvres, avide de s'agrandir et d'augmenter sa
gloire par la multiplication des \oe uvres, ou par la poursuite opini\^atre
et incessante d'une construction d'ensemble aux grandioses dimensions!'
(RS 326)
This choice, the choice that he almost unconsciously made for twenty years
or more, before coming to the realization that it was not, for him,
the true road to self-realization, was a direct consequence of the teachings 
and values that Grothendieck absorbed from his mother Hanka.
\vskip .8cm
\noindent {\bf The image of the mother}
\vskip .4cm
The image of the child, free to explore and construct and discover, without 
pressure or prejudice or fear, is invariably accompanied in Grothendieck's
mental universe by the image of the Mother, from whose womb the child
springs and to which he eternally seeks to return, symbol of the
psychic process of creation.  The passages expressing this vision
are sometimes metaphorical:
\vskip .3cm
{\it 
Dans mon travail de math\'ematicien, je vois \`a l'\oe uvre surtout ces
deux forces ou pulsions, \'egalement profondes, de nature (me semble-t-il)
diff\'erentes.  Pour \'evoquer l'une et l'autre, j'ai utilis\'e
l'image du {\bf b\^atisseur}, et celle du {\bf pionnier} ou de l'explorateur...
Ces deux pulsions qui m'apparaissaient comme ``de nature diff\'erente''
sont finalement plus proches que je ne l'aurais pens\'e.  L'une et l'autre
sont dans la nature d'une ``{\bf pulsion de contact}'', nous portant \`a
la rencontre de ``{\bf la M\`ere}'': de Celle qui incarne {\bf et} ce qui est
proche, ``connu'', {\bf et} ce qui est ``inconnu''.  M'abandonner \`a l'une
ou l'autre pulsion, c'est ``retrouver la M\`ere''...  } (RS 62)
\vskip .3cm
\noindent sometimes lyrical:
\vskip .3cm
{\it L'Univers, le Monde, voire le Cosmos, sont choses \'etrang\`eres au
fond et tr\`es lointaines.  Elles ne nous concernent pas vraiment.   Ce
n'est pas vers {\bf eux} qu'au plus profond de nous-m\^emes nous porte la
pulsion de connaissance.  Ce qui nous attire, c'est leur {\bf Incarnation}
tangible et imm\'ediate, la plus proche, la plus ``charnelle'', charg\'ee
en r\'esonances profondes et riche en myst\`ere -- Celle qui se confond
avec les origines de notre \^etre de chair, comme avec celles de notre
esp\`ece -- et Celle aussi qui de tout temps nous attend, silencieuse
et pr\^ete \`a nous accueillir, ``\`a l'autre bout du chemin''.  C'est
{\bf d'elle}, la M\`ere, de Celle qui nous a enfant\'e comme elle a enfant\'e
le Monde, que sourd la pulsion et que s'\'elancent les chemins du d\'esir --
et c'est \`a {\bf Sa} rencontre qu'ils nous portent, vers {\bf Elle}
qu'ils s'\'elancent, pour retourner sans cesse et s'ab\^\i mer en Elle...}
(RS 64)
\vskip .3cm
\noindent sometimes overtly sexual:
\vskip .3cm
{\it Pour l'amant, l'amante est la M\`ere, et son \'elan vers elle est \'elan
de retour vers le Giron dont il est n\'e -- dont toute chose est n\'ee...
C'est l'irr\'esistible \'elan de la naissance \`a rebours: {\bf retourner}
dans le {\bf Giron} accueillant de la {\bf M\`ere}...Dans la mort seulement
de l'amante et de l'amant l'un en l'autre, dans le Giron ruisselant de
la M\`ere, se pr\'epare et \'eclot {\bf naissance} dans l'amante et
dans l'amant.  L'amant se trouve re-n\'e dans l'amante-M\`ere -- entre les
cuisses puissantes de la M\`ere, reposant dans l'\'epuisement bienheureux
de l'Accouch\'ee.
%L'amante se trouve re-n\'ee d'elle-m\^eme, \`a la fois l'Accouch\'ee --
%M\`ere de l'amant et d'elle-m\^eme, et enfant nouveau-n\'e reposant aux
%c\^ot\'es de l'amant-nouveau-n\'e...Qu'il y ait ou non conception d'un
%nouvel \^etre -- l'Acte porte un fruit imm\'ediat et sans prix: l'\'etat
%d'enfance, la fra\^\i cheur d'innocence dans l'amante et dans l'amant --
%le renouvellement de l'\'etat d'enfance en eux.  C'est aussi le
%renouvellement de notre puissance de vie, de notre puissance d'aimer,
%de notre puissance cr\'eatrice -- qu'amante ou amant, en notre
%re-naissance nous puisons dans l'in\'epuisable Giron de la M\`ere.}
(Eloge 136)}

\vskip .3cm

All of them celebrate the incredible power of the creative force, and associate
it to the person of the Mother.  The role of Grothendieck's own mother
in his development of this persistent metaphor for creation is
a matter of interrogation and interpretation; there is no explicit answer,
but the question cannot be ignored. 

When Grothendieck mentions his mother Hanka in his writings, it is frequently 
to describe the negative influence she had over his self-development, by
deluding him with her powerful personality into believing and agreeing with
any number of ideas whose destruction formed one of the main ingredients of
the change he underwent later in life, when he discovered meditation.
Her domination, imposing on him the adoption of her own consciously 
held values in spite of himself, lasted for decades: `Il m'appara\^\i t
maintenant qu'une des forces derri\`ere mon attitude \'etait l'ascendant
que la forte personnalit\'e de ma m\`ere a exerc\'e sur moi pendant toute
sa vie, et pendant pr\`es de vingt ans encore apr\`es sa mort, pendant
lesquels j'ai continu\'e \`a \^etre impregn\'e des valeurs qui avaient
domin\'e sa propre vie' (RS 156).
Hanka was responsible for leading him to value only the most
masculine ({\it yang}) aspects of activity in life (her general attitude towards
others being one of `d\'edain hautain et quasiment universel' (Clef 102), 
for teaching him contempt 
for the `feminine' virtues of gentleness and kindness, and for convincing
him, even for decades after her death, that his parents were the most splendid
of parents and she herself the most heroic of mothers -- `elle continuait 
\`a se maintenir dans le mythe du grand et in\'egalable amour entre elle et
mon p\`ere, et dans celui de la m\`ere remarquable et \`a tous \'egards
exemplaire qu'elle avait \'et\'e,' (Clef 106) --
thereby psychologically barring him from access to deep truths concerning his 
own nature and development.

But there was another side to Hanka's effect on her son which must not be 
underestimated. Grothendieck's blindness to barriers and taboos of any
kind, social or professional, inside or outside of mathematics, is like 
an echo of his mother's violent and conscious lifelong rebellion against
any form of interdiction. 
She may have contributed, for many years at least, to his ignoring
the fundamental `feminine' facets of his own nature, but by her example
of absolute and voluntary rejection of all taboo and social constraint, she 
raised him in the absolute freedom from inhibition which
had such disastrous consequences on his social development, but allowed
the astounding overdevelopment of his forces of pure creativity.

The very numerous references to the relation between the mother and the
child and the connection of this relation to the creative impulse, while 
largely metaphorical, cannot be taken as entirely so; the metaphor of the
mother cannot be approached psychologically entirely independently of the
experience of the mother herself.
As difficult, as violent and conflictual and contemptuous as she may have been,
she gave him, or allowed him to preserve, the archetypal feeling of the
maternal womb as the nourishing matrix of creation, and the feeling of
being {\it whole}, of being exempt from the psychological fracture
(`coupure', `division dans la personne') caused by the 
`tabou de l'inceste qui coupe l'enfant de la m\`ere, comme il coupe la
vie de sa m\`ere la Mort, comme il coupe aussi une g\'en\'eration de celle
qui la pr\'ec\`ede' (RS 474).  Blind to the protective role of taboo in 
the preservation of a healthy collective society as he is blind to the 
protective role of fear, Grothendieck considers that total freedom from 
these constraints
is a necessary condition for the release of the creative impulse.  No matter
what harm may be done by the transgression of natural laws, it is always
good in that it leads to deeper and truer knowledge: 
\vskip .3cm
{\it Je sais qu'il y a
une substance nourrici\`ere dans tout ce qui m'arrive, que les semailles
soient de ma main ou de celle d'autrui -- il ne tient qu'\`a moi de manger
et de la voir se transformer en connaissance...Il n'y a ni amertume ni
r\'esignation en moi, ni apitoiement, en parlant des semailles et de la
r\'ecolte.  Car j'ai appris que dans la r\'ecolte m\^eme am\`ere, il y a
une chair substantielle dont il ne tient qu'\`a nous de nous nourrir.  Quand
cette substance est mang\'ee et qu'elle est devenue part de notre chair,
l'amertume a disparu, qui n'\'etait que le signe de notre r\'esistance
devant une nourriture \`a nous destin\'ee.} (RS 152)
\vskip .3cm
This passage is the key to R\'ecoltes et Semailles and the key to 
Grothendieck's message in general.  And it applies equally to his life and
to his work; an expression of vast generality, in quintessentially
Grothendieckian
language, of the trait that Demazure evoked purely mathematically, when
he described Grothendieck's approach as `turning the problem into its 
own solution'.



\vfill\eject
\noindent {\bf References}
\vskip .3cm

[RS] {\it R\'ecoltes et Semailles}, 1983-85, pagination according to .pdf 
version online

[Clef] {\it La Clef des Songes}, 1983 (?) pagination according to scanned 
version online

[Eloge] {\it Eloge de l'Inceste}, short nondistributed poetic fragment of a
longer (lost) text written by Grothendieck in the 1970's

[G] Une entrevue avec Jean Giraud \`a propos d'Alexandre Grothendieck (online)
\bye

Le point de vue et le langage des faisceaux introduit par Leray nous a amen\'e 
\`a regarder les ``espaces" et ``vari\'et\'es" en tous genres dans une 
lumi\`ere nouvelle. Ils ne touchaient pas, pourtant, \`a la notion m\^eme 
d'espace, se contentant de nous faire appr\'ehender plus finement, avec des 
yeux nouveaux, ces traditionnels "espaces", d\'ej\`a familiers \`a tous. 
Or, il s'est av\'er\'e que cette notion d'espace est inad\'equate pour rendre 
compte des ``invariants topologiques" les plus essentiels qui expriment la 
``forme" des vari\'et\'es alg\'ebriques ``abstraites" (comme celles 
auxquelles s'appliquent les conjectures de Weil), voire celle des ``sch\'emas" 
g\'en\'eraux (g\'en\'eralisant les anciennes vari\'et\'es). Pour les 
``\'epousailles" attendues, ``du nombre et de la grandeur", c'\'etait 
comme un lit d\'ecid\'ement \'etriqu\'e, o\`u l'un seulement des futurs 
conjoints (\`a savoir, l'\'epous\'ee) pouvait \`a la rigueur trouver
\`a se nicher tant bien que mal, mais jamais des deux \`a la fois ! Le 
``principe nouveau" qui restait \`a trouver, pour consommer les 
\'epousailles promises par des f\'ees propices, ce n'\'etait autre aussi que 
ce ``lit" spacieux qui manquait aux futurs \'epoux, sans que personne jusque 
l\`a s'en soit seulement aper\c cu...  

Ce ``lit \`a deux places" est apparu 
(comme par un coup de baguette magique...) avec l'id\'ee du topos. 
Cette id\'ee englobe, dans une intuition topologique commune, aussi bien les 
traditionnels espaces (topologiques), incarnant le monde de la grandeur 
continue, que les (soi-disant) ``espaces" (ou ``vari\'et\'es") des g\'eom\`etres
alg\'ebristes abstraits imp\'enitents, ainsi que d'innombrables autres types 
de structures, qui jusque l\`a avaient sembl\'e riv\'ees irr\'em\'ediablement 
au ``monde arithm\'etique" des agr\'egats ``discontinus" ou ``discrets".

C'est le point de vue des faisceaux qui a \'et\'e le guide silencieux et 
s\^ur, la clef efficace (et nullement secr\`ete), me menant sans atermoiements 
ni d\'etours vers la chambre nuptiale au vaste lit conjugal. Un lit si vaste 
en effet (telle une vaste et paisible rivi\`ere tr\`es profonde...), que

``tous les chevaux du roi

y pourraient boire ensemble..."

\noindent
-- comme nous le dit un vieil air que s\^urement tu as d\^u chanter toi aussi, 
ou du moins l'entendre chanter. Et celui qui a \'et\'e le premier \`a le 
chanter a mieux senti la beaut\'e secr\`ete et la force paisible du topos, 
qu'aucun de mes savants \'el\`eves et amis d antan...

La clef a \'et\'e la m\^eme, tant dans l'approche initiale et provisoire 
(via la notion tr\`es commode, mais non intrins\`eque du ``site"), que dans 
celle du topos. C'est l'id\'ee du topos que je voudrais essayer \`a pr\'esent de
d\'ecrire.

Consid\'erons l'ensemble form\'e de {\bf tous} les faisceaux sur un espace 
(topologique) donn\'e, ou, si on veut, cet arsenal prodigieux form\'e de 
{\bf tous} ces ``m\`etres" servant \`a l'arpenter. Nous consid\'erons cet 
``ensemble" ou ``arsenal" comme muni de sa structure la plus \'evidente, 
laquelle y appara\^\i t, si on peut dire, ``\`a vue de nez"; \`a savoir, 
une structure dite de ``cat\'egorie". (Que le lecteur non math\'ematicien 
ne se trouble pas, de ne pas conna\^\i tre le sens technique de ce terme. 
Il n'en aura nul besoin pour la suite.) C'est cette sorte de ``superstructure 
d'arpentage", appel\'ee ``cat\'egorie des faisceaux" (sur l'espace envisag\'e),
qui sera dor\'enavant consid\'er\'ee comme ``incarnant" ce qui est le plus 
essentiel \`a l'espace. C'est bien l\`a chose licite (pour le ``bon
sens mathématique"), car il se trouve qu'on peut ``reconstituer" de toutes 
pi\`eces un espace topologique en termes de cette ``cat\'egorie de faisceaux" 
(ou de cet arsenal d'arpentage) associ\'ee. (De le v\'erifier est un simple 
exercice - une fois la question pos\'ee, certes...) Il n'en faut pas plus pour 
\^etre assur\'e que (s'il nous convient pour une raison ou une autre) nous 
pouvons d\'esormais ``oublier" l'espace initial, pour ne plus retenir et ne
nous servir que de la ``cat\'egorie" (ou de l'``arsenal") associ\'ee, 
laquelle sera consid\'er\'ee comme l'incarnation la plus ad\'equate de la 
``structure topologique" (ou ``spatiale") qu'il s'agit d'exprimer.

Comme si souvent en math\'ematique, nous avons r\'eussi ici (gr\^ace \`a 
l'id\'ee cruciale de ``faisceau", ou de \``m\`etre cohomologique") \`a 
exprimer une certaine notion (celle d'``espace" en l'occurrence) en termes 
d'une autre (celle de ``cat\'egorie"). A chaque fois, la d\'ecouverte d'une 
telle traduction d'une notion (exprimant un certain type de situations) en 
termes d'une autre (correspondant \`a un autre type de situations), 
enrichit notre compr\'ehension et de l'une et de l'autre notion, par la 
confluence inattendue des intuitions sp\'ecifiques qui se rapportent soit \`a 
l'une, soit \`a l'autre. Ainsi, une situation de nature ``topologique" 
(incarn\'ee par un espace donn\'e) se trouve ici traduite par une situation de 
nature ``alg\'ebrique" (incarn\'ee par une ``catégorie"); ou, si
on veut, le ``continu" incarn\'e par l'espace, se trouve ``traduit" ou 
``exprim\'e" par la structure de cat\'egorie, de nature ``alg\'ebrique" 
(et jusque l\`a per\c cue comme \'etant de nature essentiellement 
``discontinue" ou ``discr\`ete").

Mais ici, il y a plus. La première de ces notions, celle d'espace, nous 
\'etait apparue comme une notion en quelque sorte ``maximale" -- une notion 
si g\'en\'erale d\'ej\`a, qu'on imagine mal comment en trouver encore une
extension qui reste ``raisonnable". Par contre, il se trouve que de l'autre 
c\^ot\'e du miroir, ces ``catégories" (ou ``arsenaux") sur lesquels on tombe, 
en partant d'espaces topologiques, sont de nature tr\`es particuli\`ere. 
Elles jouissent en effet d'un ensemble de propri\'et\'es fortement typ\'ees, 
qui les font s'apparenter \`a des sortes de ``pastiches" de la plus simple 
imaginable d'entre elles -- celle qu'on obtient en partant d un espace r\'eduit
\`a un seul point. Ceci dit, un ``espace nouveau style (ou topos), 
g\'en\'eralisant les espaces topologiques traditionnels, sera d\'ecrit tout 
simplement comme une ``cat\'egorie" qui, sans provenir forc\'ement d'un espace 
ordinaire, poss\`ede n\'eanmoins toutes ces bonnes propri\'et\'es 
(explicitement d\'esign\'ees une fois pour toutes, bien s\^ur) d'une
telle "catégorie de faisceaux".

Voici donc l'id\'ee nouvelle. Son apparition peut \^etre vue comme une 
cons\'equence de cette observation, quasiment enfantine \`a vrai dire, 
que ce qui compte vraiment dans un espace topologique, ce ne sont nullement
ses ``points" ou ses sous-ensembles de points, et les relations de 
proximit\'e etc. entre ceux-ci, mais que ce sont les faisceaux sur cet espace, 
et la cat\'egorie qu'ils forment. Je n'ai fait, en somme, que mener vers sa
cons\'equence ultime l'id\'ee initiale de Leray -- et ceci fait, franchir 
le pas.
\vskip .4cm
{\it Le petit enfant d\'ecouvre le monde comme il respire -- le flux et
le reflux de sa respiration lui font accueillir le monde en son \^etre
d\'elicat, et le font se projeter dans le monde qui l'accueille.  L'adulte
aussi d\'ecouvre, en ces rares instants o\`u il a oubli\'e ses peurs et
son savoir, quand il regarde les choses ou lui-m\^eme avec des yeux grands
ouverts, avides de conna\^\i tre, des yeux neufs -- des yeux d'enfant.}
\vskip .8cm
{\it Promouvoir une connaissance de la psych\'e dans une optique spirituelle
ne signifie nullement ignorer Eros ou le vilipender, comme tant de spirituels
se fond un devoir de le faire.  Bien au contraire, cela exige de lui 
reconna\^\i tre le r\^ole qui lui revient -- son r\^ole unique, crucial,
irrempla\c cable de force cr\'eatrice originelle, puisant dans les sources
m\^emes de la vie sur la terre.  Renier Eros, se couper de lui, c'est se
couper de la source de la cr\'eativit\'e en nous, et condamner l'\^etre
\`a se d\'ess\'echer, non seulement dans les pulsions profondes du corps
et dans son intelligence cr\'eatrice, mais aussi au plan spirituel[...]
(Clef 148)
\vskip .3cm
{\it C'est en 1977, l'ann\'ee qui a suivi l'entr\'ee de la m\'editation
dans ma vie et la ``re-naissance'' dont j'ai parl\'e pr\'ec\'edemment,
que j'ai d\'ecouvert avec surprise, mais sans y attacher d'abord une importance
particuli\`ere, que la pulsion de connaissance dans mon travail de 
math\'ematicien \'etait de la m\^eme nature que la pulsion amoureuse.
Les paroles et les images qui me venaient spontan\'ement, voulant
\'evoquer la pulsion de d\'ecouverte dans son essence, \'etaient paroles
et images de l'amour charnel que me soufflait Eros.  C'\'etait dans un
court texte ``En guise de Programme'', pour pr\'esenter un cour-s\'eminaire
sur l'icosa\`edre aux future auditeurs, dans l'espoir de secouer
l'apathie g\'en\'erale qui avait s\'evie l'ann\'ee pr\'ec\'edente...}
(Clef 29)
