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arXiv:2105.08799v1 [math.AG] 18 May 2021
THE NOTION OF SPACE IN GROTHENDIECK: FROM
SCHEMES TO A GEOMETRY OF FORMS
JOHN ALEXANDER CRUZ MORALES
UNIVERSIDAD NACIONAL DE COLOMBIA
Abstract. In this essay we give a general picture about the evo-
lution of Grohendieck’s ideas regarding the notion of space. Start-
ing with his fundamental work in algebraic geometry, where he
introduces schemes and toposes as generalizations of classical no-
tions of spaces, passing through tame topology and ending with
the formulation of a geometry of forms, we show how the ideas of
Grothendieck evolved from pure mathematical considerations to
physical and philosophical questions about the nature and struc-
ture of space and its mathematical models.
1. Introduction
Alexander Grothendieck is one of most influential mathematicians of
the last century and it is not difficult to argue that is one of the most
influential through the history of mathematics. The reach and depth of
Grothendieck’s work have been extensively discussed in many places,
see for instance [2, 14, 20] and the bibliography therein. Our goal is
more modest since we do not pretend to cover fully Grothendieck’s
ideas but just focus on a specific, but central part of his work, namely
the notion space1.
It is very clear, even in a superficial reading of Grothendieck’s work,
that the search of an adequate definition of the notion of space is a leit-
motif for Grothendieck, from the very beginning of his mathematical
career when he was interested in nuclear spaces to the last mathemat-
ical reflections in the 80’s of the last century when he tried to develop
1Since this text is thought for a general audience including mathematicians,
philosophers, philosophers of mathematics and, in general, anyone interested in the
history and philosophy of the notion of space in the grothendieckean work, we will
avoid the technicalities and will privilege a conceptual approach.
1
a geometry of forms as a new foundations for topology and the physi-
cal space2, passing through his monumental work in algebraic geometry.
The question What is a space? has been one of the main questions
the human being has asked since the ancient times. Many philoso-
phers, natural philosophers3and mathematicians have tried to give an
answer4and even though we have now a better understanding of the
problem we are still far from a definitive answer. The question is hard
since it involves not only mathematics but also physics and philosophy.
This was cleverly noted by Grothendieck in the 80’s5and we will return
to this point later in this text.
In fact, in [9] Grothendieck writes about the notion of space:
“La notion d’“espace” est sans doute une des plus anciennes en math´ematique.
Elle est si fondamentale dans notre appr´ehension “g´eom´etrique” du
monde, qu’elle est rest´ee plus ou moins tacite pendant plus de deux
mill´enaires. C’est au cours du si`ecle ´ecoul´e seulement que cette notion
a fini, progressivement, par se d´etacher de l’emprise tyrannique de la
perception imm´ediate (d’un seul et mˆeme “espace” qui nous entoure),
et de sa th´eorisation traditionnelle (“euclidienne”), pour acqu´erir son
autonomie et sa dynamique propres. De nos jours, elle fait partie
des quelques notions les plus universellement et les plus couramment
utilis´ees en math´ematique, famili`ere sans doute `a tout math´ematicien
sans exception. Notion protiforme d’ailleurs s’il en fut, aux cents et
2It is known that during his retirement in Lasserre since 1991 until his death,
Grothendieck wrote extensively about different topics and it would be plausible
to think that one of these topics was the notion of space. However, this is a mere
speculation, so for the purpose of this text, we do not take into account that period.
3We will use the expression natural philosopher as it was understood in 17th and
18th century.
4There is no point to make an extensive list of the people who have been in-
terested in the problem of the space but this list should include people like Plato,
Aristotle, Archimedes, Leibniz, Newton, Riemann, Einstein, just to mention some
outstanding examples. Clearly Grothendieck follows this line.
5Of course, Grothendieck was not the first one who notes this. However, we
want to emphasize that Grothendieck realized the importance of philosophy and
physics as complements of the mathematical approach to the notion of space, since
traditionally the studies on Grothendieck’s work have not paid attention to this
aspect. On the contrary, it is usually believed that Grothendieck was not interested
at all in the physical and philosophical aspects of the notion of space, and this is
partially true, if we only focus on Grothendieck’s ideas before the 80’s. Nevertheless,
his mind changed a lot, regarding physical and philosophical issues, and this is one
of the points we want to discuss in this essay.
2
mille visages, selon le type de structures qu’on incorpore `a ces espaces,
depuis les plus riches de toutes (telles les v´en´erables structures “euclidi-
ennes”, ou les structures “affines” et “projectives” , ou encore les struc-
tures “alg´ebriques” des “vari´et´es” de mˆeme nom, qui les g´en´eralisent
et qui assouplissent) jusqu’aux plus d´epouill´ees : celles o`u tout ´el´ement
d’information “quantitatif” quel qu’il soit semble disparu sans retour,
et o`u ne subsistent plus que la quintessence qualitative de la notion
de “proximit´e” ou de celle de “limite”, et la version la plus ´elusive de
l’intuition de la forme (dit “topologique”)”.
Following Deligne [7] we can think of a space as something for which
localization makes sense. In this approach the key word is localiza-
tion. Thus, the question What is a space? might be replaced by What
does localization mean?. Along this essay we will try to show how
Grothendieck approached this question. The basic idea of localization,
the one Deligne had in mind, is exemplified by the following situation6:
Let Xbe a topological space. In order to define a continuous function
f:X→R, it suffices to define it on opens subsets Uicovering X
and to check the agreement on two by two intersections. This way the
notion of continuous function is defined by localization on X.
In the example above there are two important points that play an
important role in the grothendieckean notion of space. The first one
is the absence of points, replaced by another thing, in the example by
open sets, and the second one is that those open sets cover the whole
thing. It has been remarked in other places (see [2]) how problematic
the notion of point could be in order to understand what a space is.
One of the main features in the several generalizations Grothendieck
proposes for the notion of space is that for Grothendieck the space is not
made by points. Points are just marks that can be put on the space but
not its fundamental elements. On the other hand, the idea of covering
is essential in order to understand the relation between the local and
the global aspects of the space and it also will be important in the
discussion between the continuum and the discrete in Grothendieck’s
geometry of forms as we will see later.
1.1. A brief biographical note. We will give a very short review
of Grothendieck’s life. More detailed accounts can be found in [2, 17,
6This example is due to Deligne who wrote it to the author in [7]
3
20, 13]7. Alexander Grothendieck was born in Berlin on March 28
1928. Son of Alexander Schapiro and Hanka Grothendieck both active
political militants. Due to the activities of his parents, Grothendieck
lived nearby Hamburg8under the tutelage of the family of a protestant
priest until 1939 when he met his family9. However, his father was
taken prisoner in the concentration camp Le Vernet and then deported
to Auschwitz where he died and Alexander and his mother were sent
to the camp Rieucros. This compromise of his parents with political
issues will be very influential in Grothendieck’s point of view, both
mathematical and non-mathematical.
From 1945 to 1948 Grothendieck studied mathematics at Montpellier.
In R´ecoltes et Semailles [9] he recollects the years in Montpellier and
how in solitude developed a measure theory close to the one developed
by Lebesgue many years ago. This episode marked his life as mathe-
matician. As Grothendieck wrote (see [9]) thanks to this he learnt the
importance of the solitude in the work of a mathematician. A solitude
that accompanied him his whole life beyond mathematics.
In 1949 Grothendieck arrived to Paris to Cartan’s seminar at the ´
Ecole
normale sup´erieure. This was an important turn on Grothendieck’s
mathematical education, since he moved from the border (Montpel-
lier) to the center of French mathematics (Paris). In Paris and Nancy,
where he wrote his doctoral dissertation under Jean Dieudonn´e and
Laurent Schwartz, Grothendieck could show his enormous mathemati-
cal talent. After this, he spent time as postdoctoral researcher in Sao
Paulo and Kansas.
During his time in Kansas in 1955 Grothendieck’s work took an impor-
tant turn since he left analysis and started a new mathematical road
in (algebraic) geometry. It is in this stay when he wrote the main ideas
of his work in abelian categories10 and also he started to work in his
7We include this brief note in order to give the reader an idea of Grothendieck’s
particular life and shed light on how his personality as mathematician, and more
general as human being, was formed.
8The years in Hamburg had a big impression in Grothendieck. We were informed
by Winfried Scharlau [18] that Grothendieck visited Hamburg around 2006. He
knew this by some local people that informed him on the visit. In some sense, it
seems that Grothendieck was recollecting his own steps.
9Grothendieck’s parents took part in the Spanish civil war enrolled in the anar-
chist militias.
10This work is known as Tˆohoku because of the journal where it was published.
4
version of the Riemann-Roch theorem. Thus, his monumental recon-
struction of algebraic geometry started and, even more, his quest of an
adequate definition of the notion of space really started in an explicit
way.
Since 1958 until 1970 Grothendieck occupied a position at IHES. With
Dieudonn´e as his scribe, Grothendieck wrote the giant treatise Ele-
ments of Algebraic Geometry (EGA) and ran his celebrated Seminar
of Algebraic Geometry (SGA). Grothendieck was positioned as the main
expert in algebraic geometry in the world and won the Fields medal in
1966 for his fundamental work. During this time he introduced many
important concepts (schemes, toposes among others) in order to carry
out his program of providing a vast generalization of the notion of space.
Despite the success of his seminar and the fruitful years at the IHES,
Grothendieck resigned his position in 1970 arguing ethical reasons be-
cause the founding the IHES was receiving from the military agency11.
After being involved in ecological activities in the group Survivre et
vivre he moved back to Montpellier and took one position as professor.
It is argued that Grothendieck left mathematics after his departing of
IHES since he ceased any mathematical publication but this is far from
being true. Certainly Grothendieck abandoned the establishment but
this does not mean that he abandoned mathematics. In fact, many im-
portant Grothendieck’s ideas were developed during the late 70’s and
the beginning of the 80’s when he was in Montpellier.
In order to have an idea of Grothendieck’s mathematical interests dur-
ing the 80’s, it is good to see [11]. This influential text12 contains
the main ideas of Grothendieck after his reconstruction of the alge-
braic geometry. There we can find discussions about moduli spaces of
Riemann surfaces and the Teichm¨uller tower, tame topology (another
approach to a generalization of the notion of space), children drawings,
anabelian geometry. This text condensed many of the mathematical
thoughts written in manuscripts that widely circulated between the
mathematical community.
11This is the universal accepted reason on Grothendieck’s resignation to the
IHES. However, some authors (see [16], for instance) believe that there could be
deeper reasons behind Grothendieck’s demission.
12It is known the story on how Voevodsky, being a student in Moscow, started
to learn French just to read Grothendieck’s Esquisse. Certainly this reading was
very important in Voevodsky’s career.
5
It is interesting to note, however, that some ideas concerning the geom-
etry of forms that will be important for us, in the sense that they con-
stitute an important grothendieckean approach to the notion of space
and its relations with physics and philosophy, written in an relatively
unknown manuscript [12], are not discussed at the Esquisse. These
ideas were really important for Grothendieck as we can see from this
extract of a letter sent to Tsuji on July 4th 1986:
“Excusez-moi d’avoir laiss´e passer quelques jours avant de r´epondre `a
votre lettre pr´ec´edente - depuis pr`es d’un mois je suis lanc´e sur un
nouveau th`eme math´ematique. Des nouveaux fondements de la topolo-
gie, dans un esprit tr`es diff´erent de celui de la topologie g´en´erale -
j’ai envie d’appeler cette nouvelle topologie “g´eom´etrie des formes” ou
“analysis situs”, et il est bien possible que ce sera l`a le premier texte
math´ematique que je publierai, apr`es mon d´epart de 1970 (et avant les
divers textes pr´evus, que j’annonce dans l’introduction `a R´ecoltes et
Semailles)...”
Aside from the mathematical texts, Grothendieck wrote in the 80’s
other general texts of great interest. His long text R´ecoltes et Semailles
is a jewel in the whole sense of the word. There Grothendieck not only
explored his past as mathematician but also discussed the intricate as-
pects of mathematical creativity and the ethics of a mathematician. In
La clef des songes, Grothendieck analysed his dreams and arrived to a
personal discovering of God.
In 1991 Grothendieck decided to move to Lasserre and only kept con-
tact with few people. During this time he wrote extensive texts about
different topics, but it is still a mystery the exact content of the thou-
sands of pages he wrote in his isolation. He passed away on November
13 2014 at Saint-Girons, leaving behind him a big legacy for the future
generations.
2. A walk around algebraic geometry
In this section we will discuss two generalizations of the notion of space
that Grothendieck proposed as part of his program of reconstruction
of algebraic geometry. Those generalizations are the notion of scheme
and topos. These two concepts are central in Grothendieck’s work, and
particularly relevant if one wants to understand Grothendieck’s idea of
space. In [9] Grothendieck examined his more important contributions
6
in mathematics and regarding toposes and schemes he writes:
“Parmi ces th`emes, le plus vaste par sa port´ee me paraˆıt ˆetre celui
des topos, qui fournit l’id´ee d’une synth`ese de la g´eom´etrie alg´ebrique,
de la topologie et de l’arithm´etique. Le plus vaste par l’´etendue des
developpements auxquels il a donn´e lieu d`es `a present, est le th`eme
des sch´emas... C’est lui qui fournit le cadre “par excellence” de huit
autres parmi ces th`emes envisag´es, en mˆeme temps qu’il fournit la no-
tion centrale pour un renouvellement de fond en comble de la g´eom´etrie
alg´ebrique, et du langage alg´ebrico-g´eom´etrique.”
It is clear from this quote the unifying role played by schemes and
toposes inside Grothendieck’s mathematical vision. Thus, these two
concepts allow to study not only topological or geometric aspects of
the space but also algebraic and arithmetic ones. Here, we would like
to emphasize that from the very beginning Grothendieck’s idea of space
involve a particular entity of arithmetic and geometric nature. That is
what Zalamea [20] calls the space-number13 in analogy with the space-
time in physics. We agree with Zalamea’s interpretation, so we believe
we need to understand Grothendieck’s idea of space in terms of his idea
of merging arithmetic and topology/geometry14. Therefore, schemes
and toposes can be seen as the first instances of the space-number that
Grothendieck had in mind.
2.1. Schemes: generalizing algebraic varieties. In his ICM talk
in 1958 [10] Grothendieck introduced his vast program of generaliza-
tion of algebraic geometry that took him the next decade. During his
years at the IHES he dedicated his life15 to develop these ideas. One
of the central notion he introduced is the notion of scheme and for us
this is important since it is his first generalization of the notion of space.
13In [4] we associate Zalamea’s idea of the space-number to the idea of form in
Grothendieck. We will go back to this point later in this text, since it is important
in this discussion about the notion of space.
14This is a manifestation of the duality discrete/continuum. When we discuss
Grothendieck’s ideas of the space coming from his work in the 80’s, this dichotomy
will be clearer. Now, it is interesting to notice that some mixed discrete/continuum
structure for the space is present in Grothendieck’s first works in algebraic geometry,
at least in an implicit way, and it will be more relevant when Grothendieck tries to
think in the physical space.
15Grothendieck’s capacity of work is legendary. It is said that he could spend
until 12 hours of continuous work.
7
As it is discussed in [8, 20] Grothendieck’s intrepid idea is to accept
that for every commutative ring A(with unit) it is possible to define
an affine scheme X= Spec(A). What is needed here is to define the
structure sheaf ˜
A. In this case, the structure sheaf is the sheaf of rings
over Xwith a basis of open sets {x∈X:f(x) = [f]x6= 0}with f∈A
and with fibres the local rings Axwith x∈X. Thus, considering a
ringed space, i.e. a pair (X, OX) with Xa topological space and OX
a sheaf of rings over X, we say that it is an affine scheme if it is iso-
morphic to (Spec(A),˜
A). A general scheme is defined by gluing affine
schemes.
Note that in the definition above the points of a scheme are not rele-
vant. The important thing is that a scheme is an object where we can
make localization, in the sense we discussed before. Thus, a scheme is a
space of algebraic geometric nature that generalizes the notion of alge-
braic variety. Indeed, Grothendieck characterized the notion of scheme
as a metamorphosis of the notion of algebraic variety (see [9]). The use
of the word metamorphosis is interesting, since it suggests a natural
transformation16 from the notion of algebraic variety to that of scheme.
About naturalness of the notion of scheme, Grothendieck wrote in [9]
“La notion de sch´ema est la plus naturelle, la plus“´evident” imagin-
able, pour englober en une notion unique la s´erie infinie de notions de
“vari´et´e” (alg´ebrique) qu’on maniait pr´ec´edemment ...”
And also in the same text adds
“L’id´ee mˆeme de sch´ema est d’une simplicit´e enfantine - si simple, si
humble, que personne avant moi n’avait songe `a se pencher si bas. Si
“b´ebˆete” mˆeme, pour tout dire, que pendant des ann´ees encore et en
d´epit de l’´evidence, pour beaucoup de mes savants coll`egues, ¸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 “¸ca marchai” bel
et bien...”
It is clear that the notion of scheme is for Grothendieck the natural
way in which the notion of algebraic variety should evolve, but also
he noticed that the notion was always there, but it was needed to
16Naturalness is important for Grothendieck. In his view, mathematical concepts
must be natural, in contrast to artificial and intricate constructions.
8
have the right eyes (his eyes in this case) to see it. Grothendieck was
interested in capturing in one notion the multitude of varieties that
arise when we work over prime characteristics. Again, the dichotomy
discrete/continuum is present here. This is the fundamental aporia
associated to the notion of space and Grothendieck started to struggling
with it by introducing the idea of scheme. However, the notion of space
is richer than that of algebraic variety and Grothendieck knew that.
2.2. Toposes: spaces without points. At this point we have just
discussed a restricted idea of space since we have considered only those
spaces that are algebraic varieties and look at their metamorphosis in
the idea of scheme. However, for Grothendieck the space is something
more general and, in this sense, the notion of scheme is not enough
to capture the subtleties of space. Thus, it is necessary to go beyond
algebraic geometry and entering in the realm of topology. The right
notion to start doing that is the notion of topos.
In order to understand what a topos is we need to start with the idea
of site. Here we will follow the description in [20]. Let us consider a
category Cand let us define a topology on it. For any object Xof C
it is possible to define a set of sieves J(X)17 satisfying some axioms,
namely :
1. For any object X∈ C, any sieve R∈J(X) and any morphism
f:Y→X∈ C, the sieve R×XYis in J(Y).
2. For any sieves of X,Rand R′, if R∈J(X) and for any Y→R,
R′×XY∈J(Y), then R′∈J(X).
3. For any object Xof C,X∈J(X).
A category Cendowed with a topology that satisfies 1,2 and 3 is called a
site. We focus on the idea of site since it allows us to illustrate Deligne’s
point that a space is something where localization makes sense. How-
ever, a site is just the first germ in the generalization of topological
spaces that Grothendieck had in mind. Then, what is needed to be
considered is a sheaf over a site.
17A sieve can be identified with subob jects of the dual category.
9
Thus, a topos18 is the category of all sheaves on a site. Therefore, a
topos can be seen as a generalized space. Like schemes that can be
seen as a metamorphosis of the idea of algebraic variety, a topos can
be seen as a metamorphosis of the idea of topological space. Again,
the word metamorphosis is quite accurate, since it allows to see how
toposes arise as a natural deformation of the classical idea of topologi-
cal space19.
Grothendieck introduced the idea in order to deal with cohomology
theories in algebraic geometry, and in particular, toposes play an im-
portant role in Grothendieck’s approach to Weil conjectures. This way,
toposes also arise as an important objects in problems of arithmetic na-
ture. Again, it is possible to see that for Grothendieck the notion of
space is something beyond purely geometric/topological considerations
and it is central in whole mathematics. In [9], Grothendieck points out:
“Celle de topos constitue une extension insoup¸conn´ee, pour mieux dire,
une m´etamorphose de la notion d’espace. Par l`a, elle porte la promesse
d’un renouvellement semblable de la topologie, et au del`a de celle-ci, de
la g´eom´etrie. D`es `a pr´esent d’ailleurs, elle a jou´e un rˆole crucial dans
l’essor de la g´eom´etrie nouvelle (surtout `a travers les th`emes coho-
mologiques ℓ-adique et cristallin qui en sont issus, et `a travers eux,
dans la d´emonstration des conjectures de Weil)”.
In the same text Grothendieck adds:
“C’est le th`eme du topos, et non celui des sch´emas, qui est ce “lit”, ou
cette “rivi`ere profonde”, o`u viennent s’´epouser la g´eom´etrie et l’alg`ebre,
la topologie et l’arithm´etique, la logique math´ematique et la th´eorie
des cat´egories, le monde du continu et celui des structures “discon-
tinues” ou “discr`etes”. Si le th`eme des sch´emas est comme le coeur
de la g´eom´etrie nouvelle, le th`eme du topos en est l’enveloppe, ou la
demeure. Il est ce que j’ai con“¸cu de plus vaste, pour saisir avec fi-
nesse, par un mˆeme langage riche an r´esonances g´eom´etriques, une
“essence” commune `a des situations des plus ´eloign´ees les unes des
autres, provenant de telle r´egion ou de telle autre du vaste univers des
18In this text, we use the word topos in order to refer Grothendieck topos. There
is a more general definition of topos (elementary topos) due to Lawvere but we do
not consider it here.
19We will see later that this metamorphosis does not stop in the notion of topos
and will take another forms in terms of moderate topology and geometry of the
forms.
10
choses math´ematiques.”
From the last quote is very clear that for Grothendieck the notion of
space, with its avatar in the notion of topos, is ubiquitous and central
in mathematics 20 and adequate device to study in a unified way arith-
metic and geometry. A place where discrete and continuous structures
can live together in harmony.
So far, we have seen a generalization of the notion of space running
in two steps. First one corresponding to the notion of scheme and
the second one corresponding to the notion of topos. This two steps
generalization occurs inside the domain of algebraic geometry which
was the main object of Grothendieck’s reflections between 1958 and
1970. Thus, starting with the natural spaces in algebraic geometry, i.e.
algebraic varieties, Grothendieck tried to find an archetypical structure
for the space in the notion of topos. However, as we will see below the
notion of topos does not constitute the final step in Grothendieck’s
thoughts related to the notion of space. There are other aspects to
be considered and the idea of going beyond the notion of topos will
dominate Grothendieck’s work during the 80’s.
3. Searching for new foundations: tame topology
Around the middle of the 80’s Grothendieck wrote an interesting text
[11] detailing several research programs he was thinking during the
70’s, after his retirement of the mathematical milieu, that he considered
would play an important role in the future mathematical research. One
of these programs concerns to the development of the so-called tame
topology, a new foundations for topology. In Grothendieck’s words (see
[11]):
“I would like to say a few words now about some topological considera-
tions which have made me understand the necessity of new foundations
for “geometric” topology, in a direction quite different from the notion
of topos, and actually independent of the needs of so-called “abstract”
algebraic geometry (over general base field and rings). The problem I
started from, which already began to intrigue me some fifteen years ago,
20The notion of topos is, in certain sense, marginal in mathematics nowadays.
However, recent works of Alain Connes and his collaborators (see [3], as an example)
have brought new life to the idea of topos and we believe this concept will play
a central role in major mathematical problems like the Riemann hypothesis, for
instance.
11
was that of defining a theory of “d´evissage” for stratified structures, in
order to rebuild them, via a canonical process, out of “building blocks”
canonically deduced from the original structure.”
Here, in this quote, we see how Grothendieck wanted to propose a new
setting that goes beyond the idea of topos and, in fact, beyond alge-
braic geometry. As we have seen, the origin and the development of
the notion of topos is linked to the development of algebraic geometry
in the setting Grothendieck envisioned during the 60’s, so the notion
of space developed during this period. Nevertheless, Grothendieck re-
alized that in order to capture the essence of the space is necessary to
considered a more general picture. Thus, with his idea of tame topol-
ogy, the notion of space in Grothendieck started to be independent of
the algebraic-geometric origin of Grothendieck’s first approach.
Grothendieck chose an axiomatic approach in order to understand what
a tame space could be. In his words:
“My approach toward possible foundations for a tame topology has been
an axiomatic one. Rather than declaring (which would indeed be a per-
fectly sensible thing to do) that the desired “tame spaces” are no other
than (say) Hironaka’s semianalytic spaces, and then developing in this
context the toolbox of constructions and notions which are familiar from
topology, supplemented with those which had not been developed up to
now, for that very reason, I preferred to work on extracting which ex-
actly, among the geometrical properties of the semianalytic sets in a
space Rn, make it possible to use these as local “models” for a notion
of “tame space” (here semianalytic), and what (hopefully!) makes this
notion flexible enough to use it effectively as the fundamental notion
for a “tame topology” which would express with ease the topological in-
tuition of shapes. ”
Even though, as Grothendieck wrote, a tame space is not a semianalytic
space, it is illustrative, in order to understand the idea Grothendieck
had in mind, to discuss a bit the notion of semianalytic set, since
Grothendieck did not provide an explicit definition of tame space in
[11].
Let Mbe a real analytic manifold and consider a subset Xof M, the
Xis called semianalytic if each a∈Mhas a neighbourhood Usuch
that X∩ U ∈ S(O(U)), where O(U) denotes the ring of real analytic
functions on Uand S(O(U)) denotes the smallest family of subsets of
12
Ucontaining all {f(x)>0},f∈ O(U), which is stable under finite
intersections, finite unions and complement.
Without a precise definition, but using the notion of semianalytic space
as local model, it is possible to see that the new tame spaces fit in the
general definition of space we have assumed following Deligne. How-
ever, there is an important point in the origin of the idea of tame
spaces that it did not appear in the previous notions of scheme and
topos. When Grothendieck formulated the idea of tame space he was
thinking in a right framework to deal with the notion of geometrical
shape. Thus, the main motivation here is to develop a notion of space
that allows us to study the geometric properties of shapes21. This point
is very clear from this quote (see [10]):
“After some ten years, I would now say, with hindsight, that “general
topology” was developed (during the thirties and forties) by analysts
and in order to meet the needs of analysis, not for topology per se, i.e.
the study of the topological properties of the various geometrical shapes.
That the foundations of topology are inadequate is manifest from the
very beginning, in the form of “false problems” (at least from the point
of view of the topological intuition of shapes) such as the “invariance
of domains”, even if the solution to this problem by Brouwer led him
to introduce new geometrical ideas.”
Unlike the notion of scheme and topos, both central in algebraic ge-
ometry and widely known for mathematicians nowadays, the notion
of tame space, and therefore tame topology, is marginal in mathemat-
ics. There are some developments in logic [19] in relation to O-minimal
structures, but besides this the idea of tame space is not that relevant22.
However, in our study of the notion of space in Grothendieck is, in fact,
very important, since it provides the first attempt of Grothendieck to
formulate a theory of space with the clear motivation of providing a
setting to study shapes as foundational objects for geometry/topology,
as we already mentioned. In this sense, we believe that it is not possible
21In the next section we will see that with the same motivation of understand-
ing the elusive idea of form (being geometrical shapes just one instance of this
archetypical notion) Grothendieck was beyond tame spaces and proposed the so-
called geometry of forms.
22We consider that it would be a very interesting research program to try to
develop Grothendieck’s ideas in [11] related to tame topology. We think that this
could open new geometric perspectives and new understanding of topological phe-
nomena as Grothendieck envisioned.
13
to understand how the idea of space evolved in Grothendieck’s work
without passing through the discussion about tame topology. On the
other hand, we will see that in order to tackle the problem of under-
standing shapes, this is not the final answer that Grothendieck wanted
to provide.
It is important to remark that the notion of tame space does not pre-
tend to generalise the idea of topological space (like toposes do) but pro-
vides a new understanding for it. In that sense, a tame space is a new
form that the notion of space can take, it is new type of space. Thus,
we can see that the notion of space is really protean for Grothendieck.
4. Towards a new geometry of forms
We saw in the last section how part of the motivation for developing a
tame topology was to construct a right notion of space in order to deal
with the various geometrical shapes. In fact, the notion of form23 was
one of the main subjects of Grothendieck’s reflections in the 80’s, and
in some sense tame topology was not enough for Grothendieck in order
to provide a general setting for the notion of space if we want to study
the geometric properties of forms. In this direction, Grothendieck pro-
posed a new approach for the foundations of topology. This is what he
called geometry of forms. In a letter to Yamashita on July 9th 1986,
Grothendieck wrote:
“I have been very intensely busy for about a month now, with writing
down some altogether different foundations of “topology”, starting with
the “geometrical objects” or “figures”, rather than with a set of “points”
and some kind of notion of “limit” or (equivalently) “neighbourhoods”.
Like the language of topoi (and unlike the so-called “moderate space”
theory foreshadowed in the Esquisse, still waiting for someone to take
hold of the work in store ...), it is a kind of topology “without points” -
a direct approach to “shape”. I do hope the language I have started de-
veloping will be appropriate for dealing with finite spaces, which come
23In [4] we have argued that there is an important difference between form and
shape and we have discussed how the notion of form is more general than that of
shape. In fact, Grothendieck himself chose to use the word form rather than shape,
as we can see in a letter to Ronald Brown on June 17th 1986:
“I’ve been very strongly involved with mathematics lately, working out another ap-
proach to “topology” and “form” (different from topological spaces, from topoi and
from moderate spaces as proposed in Esquisse d’un Programme), just getting the
basic language straight ...”
14
off very poorly in “general topology” (even when working with non-
Hausdorff spaces)”
Grothendieck wrote a manuscript [12] about these ideas. The manu-
script remained relatively unknown (except for a few mathematicians)
until the publication online of Grothendieck’s manuscripts by Univer-
sit´e de Montpellier. However, due to Grothendieck’s handwritten it is
not easy to decipher its content. To the best of our knowledge the first
places where there is an attempt to discuss the content of the manu-
script are [5, 6] based on some correspondence of Grothendieck where
he wrote about it. For the purpose of this essay, we consider that the
ideas on the manuscript that we can extract from the correspondence
are very important since they allow us to really understand how the
notion of space was evolving in Grothendieck’s work. We want to argue
that a full understanding of the problem of space in Grothendieck is
impossible without studying the manuscript or at least its main ideas.
However, a deep study of the text is still a work in progress, but here
we hope to give a good idea of what Grothendieck was thinking24.
In the extract of the letter to Yamashita we have quoted there is one
important point that we want to remark. Grothendieck pointed out
24Just to provide an idea of the text, we want to give the title of the different
chapters of the manuscript.
Chap 1. On a topology of the (topological) forms. Chap 2. Topological realisations
of networks. Chap 3. Networks via decompositions. Chap 4. Analysis situs (First
attempt). Chap 5. Algebra of figures. Chap 6. Analysis situs (Second attempt).
Chap 7. Analysis situs (Third attempt). Chap 8. Analysis situs (Fourth attempt).
Chap 9. Notes.
It is interesting to see how Grothendiek uses the expression analysis situs that
evokes Leibniz and Poincar´e. In a letter to Yamashita on September 16th 1986,
Grothendieck mentioned Poincar´e conjecture in relation to the analysis situs he was
invented. It is still matter of study to understand the relation between Poincar´e
and Grothendieck (and also between Grothendieck’s ideas and Leibniz’ ideas), but
we find illustrative this extract of the letter:
“I have been impressed, of course, by your second list of “newest events”. I didn’t
even know about the solution of Poincar´e’s conjecture - comes just at the right time
for me, to be able to justify within the framework of the “geometry of forms” or
“analysis situs” I am developing in terms of extant “general topology”, a certain
definition of “regular figure” (the combinatorial substitute for “variety”) I had in
mind...”.
Grothendieck refers the proof announced by Colin Rourke and Eduardo Rˆego. The
proof turned out to be incorrect. However, some years after this, Gregory Perelman
found a right proof for the conjecture. So it is tempting to ask: What is the relation
between Poincar´e conjecture and Grothendieck’s geometry of forms?
15
that his new approach to the notion of space is closer to the idea of
topos than the idea of tame space, in the sense that these new spaces
are not made by points, but also because here he is looking for a direct
approach to shapes. However, looking at the discussion in [10] con-
cerning tame topology we can see that the motivation of Grothendieck
was to understand the forms from a geometric point of view. Is there
any contradiction here? We think the answer is no.
One thing is to be motivated by some idea and another thing is that
this idea can be carried out by certain construction we propose. Our
conjecture is that Grothendieck realized that his tame topology was
not the right framework for the goal of understanding forms, taking
forms in a very broad sense, and for this reason he saw the necessity
of producing a new theory and that is what he expressed in the letter
to Yamashita. Actually, Grothendieck explored different approaches
during the years in order to understand what exactly a space is and
the idea of tame space can be thought as one of them. Before his re-
tirement at Lasserre in June 24th 1991, Grothendieck wrote a letter to
an unknown recipient25:
“Il est vrai qu’us cours de dix derni`eres ann´ees, j’ai r´efl´echi ici et l`a
`a diverses extensions de la notion d’espace, en gardant `a l’espirit la
remarque p´en´etrante de Riemann. J’en parle dans quelques lettres `a
des amis physiciens ou “relativistes”. Il ne doit pas ˆetre tr`es difficile
p. ex. de d´eveloper une sorte de calcul diff´erentiel sur des “vari´et´es”
qui seraient des ensembles finis (mais `a cardinal “tr`es grand”), ou plus
g´en´eralement discretes, visualis´es comme format une sorte de “r´eseau”
tr´es serr´e de points dans una vari´et´e C∞(p. ex. una vari´et´e rieman-
nienne) - une sorte de g´eometri´e diff´erentielle “floue”, o´u toutes les
notions num´eriques son d´efines seulement “`a ǫpr´es”, pour un ordre
d’approximation ǫdonn´e.”
This quote is particularly relevant. Three important points appear here
that are not present in the reflections about the notion of space that
Grothendieck proposed in his work in algebraic geometry and in tame
topology. First of all, Grothendieck asks for the intrinsic structure of
the space and its discrete or continuous structure. Of course, when
he discussed about toposes the relation between the discrete and the
continuum world was present, but in the sense on how toposes can put
25The unknown recipient is identified as A.Y. This letter and its relation with
the manuscript about the geometry of forms has been discussed in [5].
16
arithmetic and geometric phenomena together, but here the difference
is that the discussion on the dichotomy continuum/discrete appears in
a more foundational and structural way. Secondly, he was not only
thinking of the mathematical space but also in the physical space. It
is widely believed that Grothendieck was not interested in physics at
all, but it seems that in his last period and regarding the problem of
space he was interested in physics and, in fact, he had contact with
some physicists and discussed with them his ideas. Finally, he recog-
nise explicitly that some of his thoughts about the space came from
Riemann. We will back to the relation with Riemann later, so let us
focus on the first and the second points we have mentioned.
What was the kind of space that Grothendieck had in mind? We can
find a first answer in the letter to Yamashita on June 9th 1986. There
Grothendieck argued:
“As Riemann pointed out I believe, the mathematical continuum is a
convenient fiction for dealing with physical phenomena, and the math-
ematics of infinity are just a way of approximating (by simplification
through “idealisation”) an understanding of finite aggregates, whose
structure seems too elusive or too hopelessly intricate for a more di-
rect understanding (at least it has been so till now). Yet it may well
be that we are approaching at present the point where the continuous
models of the physical world fail to be adequate - but the physicists
are so accustomed taking for granted the conceptual superstructure of
“continuous space” worked out by the Greeks and their successors up
to Relativity and Quantum mechanics, and to confuse it with reality,
that they may well never get aware of a widening gap between concepts
and phenomena. This brings to my mind that not only the language of
(so-called “topological”) shape is to be rethought from scratch, in or-
der to be capable to account for “shape” of finite spaces, but the same
holds with vast sectors of geometry and analysis, if not all of it. Thus
differential and integral calculus, differential equations and the like, dif-
ferentiable or Riemannian varieties, the tensor calculus on it, etc - all
this quite evidently (once you start thinking about it) should make and
has to make sense within the framework of (say) finite aggregates as
well. Just think, say, of 10100 or 101000 points pretty densely located on
the unit sphere or in the unit ball, a lot more than needed for exceed-
ing by very far he accuracy of any conceivable physical measurement,
so that no physicists nor anybody whosoever (except maybe God Him-
self) could ever possible distinguish between the ambient “continuous”
space (granting it does have some kind of “physical” existence, which
17
I doubt...) and this kind of network thought of as an “approximation”
(whereas in reality, the opposite seems to me more likely to be true:
the continuum is the concept approximating the elusive finite - but very
large - aggregate). This would mean rethinking too the notion of differ-
entiability (say) and of differential, in wholly new terms, when “passing
to the limit” means now a finite process, namely passing to “small” val-
ues of the parameter or parameters, where the rate of “smallness” is
precisely prescribed and should by all means not be exceeded, and where
the “margin of errors” admitted in the computation or definition of the
“differential” or “derivative” is equally precisely prescribed, in keeping
with the former prescription.”
Grothendieck was inclined for a discrete structure for the space and
even he thought of the continuum as a convenient fiction. This is actu-
ally quite impressive. Nobody before Grothendieck proposed to think
the continuum as an approximation to the discrete26. We find this idea
very suggestive on its own27. For Grothendieck, the continuous struc-
ture of the space can be thought as an artificial, but necessary and
convenient, construction in order to understand its actual structure.
However, the relation between the discrete and the continuum is not
that simple for Grothendieck. In fact, in [9] he proposed that there
could be three possibilities for the structure of the space. The space
can be discrete, continuum or mixed discrete/continuum structure. In
Grothendieck’s words:
“Les d´eveloppements en math´ematique des derni`eres d´ecennies ont d’ailleurs
montr´e une symbiose bien plus intime entre structures continues et dis-
continues, qu’un ne l’imaginait encore dans la premi`ere moiti´e de ce
si`ecle. Toujours est-il que de trouver un mod`ele “satisfaisant” (ou, au
besoin, un ensemble de tels mod`eles, se “raccordant” de fa“¸con aussi
satisfaisante que possible. . .), que celui-ci soit “continu”, “discret” ou
de nature “mixte” — un tel travail mettra en jeu sˆurement une grande
imagination conceptuelle, et un flair consomm´e pour appr´ehender et
26In the letter Grothendieck attributes this idea to Riemann. However, it is not
that clear that actually Riemann was thinking the continuum in this terms. We
will back to this point later.
27We discuss about this in [5]. Let us just mention that this idea is an inversion
respect to the usual way of thinking the relation between the continuum and the
discrete. Usually the continuum is seen approximate by the discrete via a saturation
(limit) procedure. Grothendieck is proposing a radical, but we think very fruitful,
change of perspective. It would be very interesting to explore in more detail this
line of thought.
18
mettre `a jour des structures math´ematiques de type nouveau. Ce genre
d’imagination ou de “flair” me semble chose rare, non seulement parmi
les physiciens (o`u Einstein et Schr¨odinger semblent avoir´et´e parmi les
rares exceptions), mais mˆeme parmi les math´ematiciens (et l`a je parle
en pleine connaissance de cause).”
Again, it is clear, that he was thinking not just in the mathematical
space but also in the physical one. Unlike what he did in algebraic
geometry and in tame topology, his new geometry of forms is not just
a mathematical description of the space but it should be thought as a
description useful for physical purposes. This is the reason, we believe,
that the notion of form was so central in this new approach. A form
has many manifestations in the world not only in mathematics, so any
theory about the space focused on the idea of form should reflect those
manifestations. We think Grothendieck had fully conscience about this.
4.1. Riemann’s heritage? An important point to analyse in Grothendieck’s
conception of the geometry of forms is Riemann’s influence. Of course,
Riemann has a big influence in the whole grothendieckean work, but
in relation to the geometry of forms that influence is particularly in-
teresting. In [9] we find:
“Il doit y avoir d´ej`a quinze ou vingt ans, en feuilletant le modeste vol-
ume constituant l’oeuvre compl`ete de Riemann, j’avais ´et´e frapp´e par
une remarque de lui “en passant”. Il y fait observer qu’il se pour-
rait bien que la structure ultime de l’espace soit “discr`ete”, et que les
repr´esentations “continues” que nous nous en faisons constituent peut-
ˆetre une simplification (excessive peut-ˆetre, `a la longue. . .) d’une
r´ealit´e plus complexe; que pour l’esprit humain, “le continu” ´etait plus
ais´e `a saisir que “le discontinu”, et qu’il nous sert, par suite, comme un
“approximation” pour appr´ehender le discontinu. C’est l`a une remar-
que d’une p´en´etration surprenante dans la bouche d’un math´ematicien,
`a un moment o`u le mod`ele euclidien de l’espace physique n’avait ja-
mais encore ´et´e mis en cause; au sens strictement logique, c’est plutˆot
le discontinu qui, traditionnellement, a servi comme mode d’approche
technique vers le continu.”
Here Grothendieck again refers Riemann as the one who was the first
in thinking the continuum as an approximation of the discrete, and in
this sense the continuous structure of the space as an approximation
19
of a more intricate discrete structure of it28. What it is also important
to consider is that Grothendieck pointed out that the relation between
the continuum and the discrete can be determined by the fact that
the continuum is easier to assimilate by the human mind (he uses the
expression human spirit) that the discrete. This has deep philosoph-
ical implications in terms of the nature of space. The question is to
some extent this idea is already in Riemann in the explicit way that
Grothendieck is presenting it.
Riemmann’s contribution started to be clarified in a letter to Yamashita
on September 16th 1986. There Grotehndieck wrote:
“When I wrote about the theory of mathematical continuum begin a
kind of approximation and idealization of very large and complex dis-
crete aggregates, I thought I was citing an idea of Riemann, from what
I remembered from reading through his collected works, many years ago
(presumably in his exposition of the idea of what is now called riemann-
ian geometry). When I later looked for a precise reference in his work,
I was rather surprised that I wasn’t able to find it - and it seems you
didn’t find it either. So maybe, I found his idea only “in between the
lines” of his actual words. ”
The “in between the lines” at the end is really remarkable. At the end
of his Habilitation thesis [15], Riemann discussed about the possibility
of a discrete structure for the space. That idea yielded a big impression
on Grothendieck. After looking with more detail, he recognised that
in an explicit way the idea is not in Riemann but he certainly believed
that it can be extracted from what Riemann wrote. This is a risky
implication. Thus, Why does Grothendieck make that implication?
We could find a partial answer again in the letter to Yamashita from
September 16th:
“It would almost seem that there has been something like “telepathic”
communication (to give it a name) from Riemann’s thought-world to
mine. I am still convinced now, or rather, I well know, that I never
discovered myself this idea (because this kind of discovery of some sig-
nificant view, I mean the moment of such discovery, one never forgets
I believe) but that I got it from Riemann, and with some surprise...”
28We already saw that Grothendieck also mentions Riemann in relation with this
idea in his letter to Yamashita on June 9th 1986
20
This passage is very beautiful. That “telepathic communication” be-
tween Grothendieck and Riemann, two of the biggest mathematical
geniuses of all times, connected by their mathematical passion and the
wish of understanding the notion of space, is one of the most remark-
able examples of mathematical creativity. Despite of being separated
by many years, both of them had the intention of unveiling the inner
and deep structure of the space and they walked together going behind
that goal. Thus, without doubt Grothedieck’s ideas about the space,
and in particular the geometry of forms he envisioned in the late 80’s,
is Riemann’s heritage.
5. The notion of space: mathematics, physics and
philosophy
In the previous section we discussed how thanks to the formulation of
his geometry of forms Grothendieck began interested in the relations
between the physical and the mathematical space. It is also impor-
tant to remark that in fact Grothedieck was beyond mathematics and
physics in his reflections and also consider the importance of philosophy
in order to understand what a space is. This give us a new perspective
about Grothedieck’s thoughts in the late 80’s and also about his under-
standing of the notion of space as a whole. In particular, Grothendieck
was interested in the role of mathematical models for the physical re-
ality and the philosophical background of them. He thought that only
putting together philosophy, physics and mathematics we can find an
adequate approach to the problem of space. Regarding this, in [9] he
wrote:
“Pour r´esumer, je pr´evois que le renouvellement attendu (s’il doit en-
core venir. . .) viendra, plutˆot d’un math´ematicien dans l’ˆame, bien
inform´e des grands probl`emes de la physique, que d’un physicien, Mais
surtout, il y faudra un homme ayant “l’ouverture philosophique” pour
saisir le noeud du probl`eme. Celui-ci n’est nullement de nature tech-
nique, mais bien un probl`eme fondamental de “philosophie de la na-
ture”.”
And in a letter to AY on June 24th 1991 he also pointed out:
““Je suspecte que les nouvelles structures `a d´egager seront beaucoup
plus subtiles qu’un simple paraphrase de mod`eles continus connus en
termes discretes. Et surtout, qu‘avant toute tentaive de d´egager des
nouveaux mod`eles, pr´esm´es meilleurs que les anciens, il s´ımpose de
21
poursuivre une r´eflexion philosophico-math´ematique tr`es servie sur la
notion mˆeme de “mod`ele math´ematique” de quelque aspect de la realit´e
- sur son rˆole, son utilit´e, et sus limites.”
It is beautiful to see how Grothendieck uses the expression natural phi-
losophy. The problem of space is not just a mathematical, physical of
philosophical one, it is a problem of natural philosophy in a very broad
sense that includes mathematics, physics and philosophy but also goes
beyond them. This is evidence of how deep Grothendieck’s thoughts
about the space became. The philosophical side of Grothendieck’s ap-
proach to the notion of space is, in general, unknown but for us it is
very relevant and it should be studied in more detail. Here, we just
mention this aspect to motivate the study and the debate about it.
Grothendieck pointed out that the problem of space is not only of tech-
nical nature. It is not enough to think of the problem in mathematical
or physical terms. We also need a right conceptual approach and it
is in this point where philosophy should enter. One of the aspects to
be consider is how the mathematical models of the space that we can
propose reflect the nature and the structure of the physical space and
it is consistent philosophically. Of course, this is not an easy problem
and Grothendieck had no answer. Again, the reflection on the role of
mathematical models is of philosophical nature as Grothendieck saw
(see [9]):
“J’ai le sentiment que la r´eflexion fondamentale qui attend d’ˆetre en-
treprise, aura `a se placer sur deux niveaux diff´erents.
1. Une r´eflexion de nature “philosophique”, sur la notion mˆeme de
“mod`ele math´ematique” pour une portion de la r´ealit´e. Depuis les
succ`es de la th´eorie newtonienne, c’est devenu un axiome tacite du
physicien qu’il existe un mod`ele math´ematique (voire mˆeme, un mod`ele
unique, ou “le” mod`ele) pour exprimer la r´ealit´e physique de fa¸con par-
faite, sans “d’ecollement” ni bavure. Ce consensus, qui fait loi depuis
plus de deux si`ecles, est comme une sorte de vestige fossile de la vi-
vante vision d’un Pythagore que “Tout est nombre”. Peut-ˆetre est-
ce l`a le nouveau “cercle invisible”, qui a remplac´e les anciens cercles
m´etaphysiques pour limiter l’Univers du physicien (alors que la race des
“philosophes de la nature” semble d´efinitivement ´eteinte, supplant´ee
haut-la-main par celle des ordinateurs. . .). Pour peu qu’un veuille
bien s’y arrˆeter ne fut-ce qu’un instant, il est bien clair pourtant que la
validit´e de ce consensus-l`a n’a rien d’´evident. Il y a mˆeme des raisons
22
philosophiques tr`es s´erieuses, qui conduisent ‘a le mettre en doute a
priori, ou du moins,`a pr´evoir `a sa validit´e des limites tr`es strictes. Ce
serait le moment ou jamais de soumettre cet axiome `a une critique
serr´ee, et peut-ˆetre mˆeme, de “d’emontrer”, au del`a de tout doute pos-
sible, qu’il n’est pas fond´e : qu’il n’existe pas de mod`ele math´ematique
rigoureux unique, rendant compte de l’ensemble des ph´enom`enes dits
“physiques” r´epertori´es jusqu’`a pr´esent.
Une fois cern´ee de fa¸con satisfaisante la notion mˆeme de “mod`ele math-
ematique”, et celle de la “validite” d’un tel mod`ele (dans la limite de
telles “marges d’erreur” admises dans les mesures faites), la question
d’une “th´eorie unitaire” ou tout au moins celle d’un “mod`ele opti-
mum” (en un sens `a pr´eciser) se trouvera enfin clairement pos´ee. En
mˆeme temps, on aura sans doute une id´ee plus claire aussi du degr´e
d’arbitraire qui est attach´e (par n´ecessit´e, peut-ˆetre) au choix d’un tel
mod`ele.
2. C’est apr‘es une telle r´eflexion seulement, il me semble, que la ques-
tion “technique” de d´egager un mod`ele explicite, plus satisfaisant que
ses devanciers, prend tout son sens. Ce serait le moment alors, peut-
ˆetre, de se d´egager d’un deuxi`eme axiome tacite du physicien, remon-
tant `a l’antiquit´e, lui, et profond´ement ancr´e dans notre mode de per-
ception mˆeme de l’espace : c’est celui de la nature continue de l’espace
et du temps (ou de l’espace-temps), du “lieu” donc o`u se d´eroulent les
“ph´enom`enes physiques”.”
We include this extensive quote because it illustrates in Grothendieck’s
own words what we are trying to argue, i.e., that the technical de-
scription of the space only makes sense for Grothendieck after a con-
ceptual approach. It took him around 40 years to reach this con-
clusion. This quote also shows the evolution of Grothendieck’s ideas
from his pure technical questions in his work in algebraic geometry
to the more physical-philosophical questions in his geometry of forms.
Thanks to this, it is very clear that notion of space is indeed a leitmotif
in Grothedieck’s mathematical life.
6. Final remarks
Along this essay we have shown how the idea of space in Grothedieck
evolved from his generalization of the notion of (algebraic) variety via
schemes to the formulation of a new approach for the foundations of
23
topology in what he called geometry of forms, passing through the no-
tion of topos and tame space. Of course, for Grothendieck the notion
of variety and topological space (in the classical sense) were useful in
mathematics but he realized of new situations where new ideas on space
were needed. In particular, one of these situations is related to the no-
tion of form and its role in topology and, in general, in mathematics.
Grothendieck wrote that there are three main areas of mathematical
reflection, namely arithmetic (associated to the notion of integer num-
bers), analysis (associated to the notion of magnitude) and geometry
(associated to the notion of form). From the beginning he put the no-
tion of form as belonging to the realm of geometry and more general
topology, but even more, it is the central notion for geometry in the
sense that geometry for him is the branch of mathematics in charge of
studying the aspects of the universe, not just the mathematical uni-
verse of course, related to the form.
Following that line, for Grothendieck there are three different notions
which are important from a mathematical point of view, namely num-
ber, magnitude and form. In [4] we argue that number and magnitude
are manifestations of a more general notion of form and in this sense
number can be thought as an arithmetical form while magnitude can
be thought as an analytic form. What Grothendieck called form should
be interpreted as a geometrical form (or shape). What we have is a fun-
damental triad that gives the most important aspects to be considered
in a mathematical approach to the notion of space, i.e. arithmetical,
analytical and geometrical aspects. We have argued that this was the
quest of Grothendieck for more than 40 years.
On the other hand, as we have already mentioned before in this text,
Zalamea’s interpretation [20] of the space-number in Grothendieck’s
work as analogue to the space-time in physics gives a nice way to un-
derstand Grothendieck’s ideas. Zalamea proposes that we should think
Grothendieck’s work as an attempt to develop the notion of space-
number as the conjunction of arithmetic and geometry in one entity.
This analogy is very suggestive since, as we have shown, Grothendieck
was interested in physical aspects of the space. For this reason, we
find very plausible to think of his last works in terms of constructing a
theory for a space-number. We think that this line of thought could be
very helpful in order to understand the notion of space in Grothendieck
and it is a future project try to follow it.
24
In [1] Bolzano presents an idea of mathematics as a general theory of
forms. In some sense Grothendieck’s approach to the notion of space is
a realization of Bolzano’s idea, including physics and philosophy in the
picture. We believe this should be a way to think of Grothendieck’s
contribution to the human knowledge. There are still many problems
to be considered and many gaps to be filled in order to understand
Grothendieck’s ideas about the space. In this text we wanted to present
a new perspective and contribute with some elements which, to the best
of our knowledge, were not consider before in the literature. More than
give a definitive answer about the problem of space in Grothendieck’s
work we wanted to give a picture, as complete as we could do it, in
order to start the dialogue. A dialogue that following Grothendieck’s
own advice will require a mathematical, physical and philosophical way
of thinking.
Acknowledgements The author wants to thank Fernando Zalamea
for many illuminating discussions on Grothendieck’s work over the
years, his constant and warm support and having read a draft ver-
sion of this essay. Of course, any mistake that remains is completely
responsibility of the author. He also thanks Winfried Scharlau for pro-
viding him with Grothendieck’s correspondence with Yamashita and
Tsuji and for conversations on Grothendieck’s life, and Yuri Manin
for supporting the idea of writing this text. Finally, he thanks Pierre
Deligne for many interesting emails and his patience in explaining his
ideas.
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