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A
nostalgia of the triangles era
- N:... Do you remember
your early attempts in researches?
- T: Yes, I translated all known theorems
in geometry R3 to geometry R4 That was, if I dare
to say, my first attempt to do something a
little bit original; but it was for me a way to
succeed in the understanding of how was made, let
us say a system of two plans in R4 Etc. and I
believe that I attained a very good intuition in
that time, and I could already see a space in four
dimensions when I was ten, or eleven years old.
- N: And so , have you other
remembrances of that period?
- T: I believe it is about the only thing
which I still remember . And also, the recollection
of a somewhat intellectual scandal felt when
my professor of fifth grade said that one could
calculate the number Pi . Such an idea that Pi
could be computed through theoretical methods, was
something which, at that time seemed to me
extremely mysterious and fascinating.
- N: Yes, why?
- T: It was usual to measure Pi with
threads around cylindrical cans, isn't it, and the
idea that there were theoretical processes allowing
this calculation was something radically new to me.
That seems to you quite a commonplace, but for me,
it was not...
Moreover, I believe that it was in third grade,
we had elementary Euclid geometry ; my teacher was
not particularly brilliant , but he had managed
to arouse my interest and I did really liked that
a lot , I resolved very complicated problems
dealing with construction of triangles, etc.
and it is a little bit , due to the nostalgia of
that time that I defend the elementary Euclid
geometry against modernists. I think, as far as I
am concerned , that if we persist in the present
direction, we will deprive ourselves of a method of
selection which was really excellent and I would
not be amazed to notice definitely in the years to
come, a real decline of the level of mathematics in
France following the renunciation of the Euclidian
geometry; that would not be surprising..
- N: You spoke about nostalgia of this
period, what did this period mean to you?
- T:... Let us say that I was somewhat
a freshman, with a sort of will to reach the
limits of the possibilities of the mind...
The idea that there was not a problem that I
could not resolve... Later on obviously one we
water our wine!... But it was the idea that there
was no problem which I could not overcome in the
area of geometry.
- N: Was it not the same in other
matters?
- T: No, as you know the algebra has
never been of a big interest for me.
- N: And at that time you already
realised that there was a difference between the
two?
- T: Oh! Yes, naturally, the analytical
geometry, from the moment you practice it, seemed
to me a good technique, but is not particularly
inspiring , whereas a problem of geometry is really
something completely... special, a lot more
enigmatic.
- N: Enigmatic?
- T: Oh Yes ! A problem of geometry is
somewhat enigmatic. In other words, in geometry,
there is no heuristics, isn't it, it is
necessary to resume everything from zero
according to the problem involved, the contrary of
what takes place in algebra
A
mathematician's vocation
... It is all about what I can tell on my
mathematician's vocation, as you see it is very
short of supply! As for the first theorem which I
demonstrated on my own , if I dare say, I believe
that it was the equivalence of the bifocal
definition and the definition by focal and
direction of the conics, by a method based upon
elementary geometry; I had submitted it to my
teacher who thought that it was already known,
which was very likely. Due to a traditional method
it was a little bit heavy, and did not please me.
The passage of the unifocal definition to the
bifocal definition is something rather
mysterious and I achieved it through a
construction which I still remember very well
to-day...
- N: Do you , remember when you said
to yourself: I want to involve myself to
mathematics?
- T: Good... The paradox thing is
that, actually I never wanted to deal with
mathematics. When I arrived at the "Ecole
Normale" ( teachers' training college), I explained
to the then assistant head-master who was Georges
BRUHAT, that obviously I was entered as a
mathematician but rather interested in the
philosophy of Sciences, as practiced CAVAILLES and
others during that period... he raised his arms to
the skies and said: please , dont do that, please ,
graduate at once and you do not bother about
philosophy of sciences! And I think that in a
certain sense, he was right; one should bother of
the philosophy only when one has assured one's
living through more classical and usual methods .
Therefore, I engaged myself to mathematics. At the
teachers' training college, I essentially attended
the seminar of CARTAN which taught a lot of things
and... in 1946, I could join the C.N.R.S. following
CARTAN to Strasbourg for one year or two. CARTAN
returned to Paris, but I remained in Strasbourg
because I enjoyed it. It is especially during the
seminar of ERESSMANN that I really learnt about the
new topology, this topology which was built up i at
that time. Years 45 to 50 were extraordinary for
the algebraic topology because an enormous quantity
of new" beings" were discovered, new techniques,
etc. " cohomology", "fibrés", " homotopy".
And it is in such a loaded stream that I wrote my
thesis which took me moreover quite a number of
years because it was over only in 1951. I would be
ready to say... (Maybe we illusion ourselves , do
we not?) but I would be tempted to say that I do
not really consider myself as what is so called a
top mathematician, in the sense that I have no
taste for the mathematical structure as such.
When I watch my colleagues, I do not want to quote
names , but examples teem all around me, they
savour beautiful structures, rich, sophisticated
structures, in which one can do heaps of things,
clarify the relations to this or to that: I am
not a lot personally, tempted by this kind of
approaches ... I am neither an ultra generalist
as my colleague GROTHENDIECK is...
Two
types of mathematicians
An American colleague, whose name I shall keep
silent here, says that there are two types of
mathematicians : the mathematician who drills very
deep wells to find there the gemstone, the precious
stone that he will study at leisure and of which he
will expose all the beauties, and the other one ,
the bulldozer which erases all the area.
Indeed, if we accept this vision of the
mathematicians, I am none of both, then perhaps I
am not at all a mathematician from this point of
view...
- N: How do you consider yourself by
the way?
- T: Oh! I don't know, let us say that
what interested me in mathematics were rather
general properties, more than the study of specific
structures... But not at all with the systematic
approach of GROTHENDIECK for example.
- N: Neither bulldozer nor
hole-driller
- T.-Neither bulldozer, nor hole-driller
(laughters)... No, I think that my success in
mathematics owes a lot to historic circumstances :
I wrote my thesis during a period when effectively
there was a whole new material, a time rather
extraordinarily blooming. I took advantage of that
movement, but afterward, I worked on things more
oriented to analysis, a theory of " applications",
" stratified ensembles", but to my opinion it is
more technical and I am sure that for the most part
of the mathematicians, it seems less interesting ,
although in some respect ,I believe that it should
be more important ...
The
catastrophe theory
- N:Did you give personally the name
of "catastrophe theory" to your works?
- T: Not exactly when I mean in my book,
the notion of regular point opposed to
disaster.
- N: But you did introduce the word of
catastrophe….
- T: Yes , I introduced the word
catastrophe in a rather specific meaning.
- N: How did this word occurred to you
?
- T: Simply because I wanted to express
the idea of a fundamental distinction, the
distinction that topologues enters in open and
closed concept .The open one means, if you wish,
something as a static , a regular state ,something
like a in sitiu balance between included dynamics ,
whereas the closed one , expresses a place of
points where something happens, a discontinuity.
Therefore, I came to the idea that closed
situations , mostly general are not very
interesting, but there is closed ones , more
regular in a sense ,which appear in a almost
unavoidable manner ... If one formulates hypotheses
on what one could call the prevailing dynamics, it
is in some sort the generalisation of the idea
of defect in physics. In an ordered environment
, as a crystal structure, there is a regular one
which sometimes bumps on certain sub-varieties so
called the mistakes; it is somewhat the same
idea.
Therefore I wish to express the idea that there
were exceptional sub-ensembles which were
associated to irregularities of dynamics and
it is why I called that catastrophes; I would have
been able indeed to take a much more neutral
terminology, which would have avoided many troubles
to me...
- N: But you have chosen such a
word
- T: I chose it in the meaning of what I
spoke about "points of disaster" oppose to regular
points; the natural opposite of regular points, are
obviously singulars ones, but the point of
'catastrophe" is still different, it is as a rule
different from a singular point...
- N: What does mean a catastrophe for
you?
- T: Let us suppose that I have a
space in which occurences happen:. I look what
is taking place and I split points into two
categories: the regular points where there is
nothing to notice at first sight, that means that
all the observable items are continuous in this
point or on the contrary, there is one which is
discontinued : therefore there is at least an
observable item which is intermittent. There is
observable discontinuity in this point, then in
that case I say that it is a point of
"catastrophe", that's all... Then why adopting such
a word? I could obviously speak simply of
discontinuity (I was blamed for that afterward) but
I wanted to promote the idea of an underlying
dynamics, a pervading dynamics which engenders
the sub-ensemble of disasters and it is for that
that I introduced this word , which moreover, had
been already used by the physicists in an
acceptance not completely similar, but neutral at
least; The physicists already spoke, in the quantum
theory of fields, of the infrared disaster, of
ultraviolet disaster. There was "catastrophes"
whic never killed anybody, I wrote it!
- N:Is something underlying would
appear...
- T: That's right, yes, finally, the very
type of "catastrophe", as you may want; let us say,
it is like a sheet of paper you are folding and
which, at a given moment, catches an angle, does it
not?; which remains regular and then suddenly is
folded , a fold characterized by a discontinuity.
It is this sort of phenomenon that I wanted to
systematize.
What
the mathematics are?
- N: What mathematics are for
you?
- T: Oh! They represent essentially
the universal theoretical language. It is to
my opinion, the only rigorous possibilities of
reaching a universally valid thought owing to
mathematics or mathematical laws; in other words, I
do not think that one can, in the sciences, have a
theorization with a really lawful universal
validity based only on concepts expressed by words
out of the common language; if these concepts are
not capable of expressing themselves mathematically
in term of fundamental entities as space and time;
what is the case in physics, isn't it?
In physics, the concepts can be expressed
mathematically out of data of space and time, of
"spatio-temporal" data. Concepts which do not allow
this kind of sumarisation will be always suspected
and the hope of the theory of the
"catastrophe", is precisely that there are in
the abstract univers sorts of germs of local
analyticity around of which one can make a kind of
mathematical "theorization" . There lays the
hope in something as an universal analytical
structure in which we may work, as it is the case
in physics.
In physics there is an universal analytical
structure, due to the group of "invariances" of
physics: group of LORENTZ, group of GALILEE, etc.
and these groups allow in a sense to make
accesible to everybody, the whole universe
because they act transitively this way, there is
a sort of universal stand-still with which
to operate, or practice quantitative mathematics; I
do not think that this situation can be generalized
in other subjects, but it can raise hope
that there is locally, in a sense the semantic
universes in which work certain concepts,
situations with character locally analytics
allowing the expression of interesting situations
of universal character; it is if you want, the
underlying philosophy in the theory of the
catastrophe.
- N: In other words, it is chiefly
this universal character which may interest
you.
- T: Yes, of course, obviously.
Reality
is mathematics
- N: I shall compare that to what you
told me right no: when you were a school-boy, you
were already thinking that it was possible to
resolve all problems.
- T: Yes, yes, it is obvious ,moreover I
wrote about it : there is only theorization in
mathematics. From this point of view, I am an
mathematical imperialist , I am blamed for that
by other disciplines... You doubtless heard about
current controversies on the theory of the
disasters? I think that people did not realize the
subversive side of this theory. The day
will come when they realise , we can expect still
stronger resistances because, fundamentally, the
mathematics, compared to other disciplines,
accepted a role of pure routine.
You have mathematicians in laboratories of
biology , even in laboratories of social sciences,
they are asked to make statistics, that's all. But
obviously the local specialist , steers all
operations; mathematics are seen only in a
ancillary role in other sciences: experimental or
so called human sciences.
- N: A device...
- T: Yes, as a tool and , personally,
I think that it is an abnormal situation and
that mathematics cleanly included can serve as
theoretical guide in a great number of
disciplines. It is in this sense I believe that
mathematics have a very great future in the
"mathématisation "of sciences,
mathématisation which will not be similar to
the model of physics, with results maybe sketchier
and softer than in physics, but no lesser interest
...
- N: Are mathematics still something
else for you?
- T: As far as it is an universal way of
thinking it is also an access to reality; in
other words, for me, the ontology is (as far as I
have a metaphysical approach, what remains
obviously a matter to dig) rather Platonic or
pythagorician; and in this way, I think that the
ground of everything in the world is mathematical
even where apparently it does not fit
- N: Is reality mathematical?
- T: I think that we may say that reality is
mathematical,… yes. But may be it is not
mathematics we know, it will obviously be necessary
to engage rather considerable extensions with
regard to the known mathematics to build relevant
mathematics aimed at biology, psychology or
sciences of this sort...
The
periods of possession
- N: When you are at your desk,
,dealing with mathematics, what is your
feeling?
- T: Well! I confess that, for quite a
lot of years, I deal no more with mathematics
stricto sensu .Sometimes I am still interested in
problems of mathematics, but that becomes more and
more rare. I was interested in peripheral
disciplines, as biology, linguistic a lot and now
geology. I rather dedicate my wilful activity to
these experimental disciplines rather than to
bother of purely called mathematics . Therefore if
I sometimes practice mathmatics, it is rather for
professional obligations than for other purpose;
but obviously, it is a rather recent evolution, for
the last ten years
Anyway, it is well known that after 35 years of
activity a mathematician cannot produce something
worthwhile, and custom, the traditional belief are
I guess widely based, so in such conditions it is
better to be involved in other things than
mathematics!
- N: But do you remember what your
life was during that period of time?
- T: Of course! Yes, I also knew these
periods of possession by a problem, naturally I
knew such situations. I knew some periods like that
in my life, but finally not many.
- N: Periods of possession?
- T: Yes, periods when a problem
monopolises you so much that we become almost
unable to think to whatever else... But as I
already told you it has been very, very rare in my
case...
A
period of crisis
- N: It is not possible any
more...
- T: Maybe that it is not possible any
more, yes; I have not enough interest for purely
mathematical problems to make me monopolised by
them. I think that most of mathematicians know in
their life a moment of crisis when they are in
doubt about the value of what they did. Especially
in front of the rising" infertility "which arrives
along with age, it is very difficult to avoid this
kind of crisis... I reacted taking interest
in other matters beyond mathematics; I think it
is not a bad method.
-N: Is it really a crisis ?
- T: Yes it looks like a little a
crisis, I think. Finally, I do not know if one
can draw general laws about it, but that seems a
little a crisis, yes. As far as I am concerned,
this crisis took place around the years 58-60.
Definitely, I believe that i is the same in
mathematics as in other disciplines and it is the
same situation as the one that EINSTEIN described
to VALÉRY. EINSTEIN had paid a visit to
VALÉRY, or VALÉRY had invited him and
then, obviously, always very anxious to understand
the mechanisms of the relativity, VALÉRY
asked heaps of questions to EINSTEIN and, in
particular, he asked him;" but finally, Master, you
get up during the night to take note of your ideas
on a small pad?" And EINSTEIN dropped: "Ideas ?Oh
You know , you get two or three in your life !"
Indeed! It is also a little my feeling , about
my mathematical work. I believe that I had two
or three ideas in mathematics and the remaining is
just technical elaboration... And yet, among these
ideas, there are some which were almost
obvious...
An
aversion to get involved in certain sectors of
mathematics
- N: Are you not proud of
yourself?
- T: Yes, of course, some works may give
you a feeling of pride, that is possible. I suppose
that MESSRS. FEIT and THOMSON, when they
demonstrated that any group of odd order is
solvable drew a legitimate pride out of it...
But there to go back to the emotional aspects of
mathematics, I believe that what is worth, is
the quasi-emotional reaction of the mathematician
towards certain theories. There are
mathematical theories which I have never been able
to imply myself because I had something like a
sort of aversion to start with and I never
could overcome it afterward, I think for example to
the theory of the groups of DREGS; also the main
part of the "functional analysis" , a branch of
mathematics to which I feel profondly reluctant
. What theories I could quote again? The
algebra, very very abstract algebra, the type of
non-commutative algebra, that too does not mean
much to me .
- N: What do you feel at such
moment?
- T: I have the impression that to get
implied, it would be at first necessary at that I
work, I am lazy, then, it would be necessary
that I understand better the motivation,
isn't it? Generally, most of these theories do not
seem to me motivated enough: I think that
it is where the core of the problem is, perhaps
it is matter of pedagogy. If one had been able to
uncover a good pedagogy related to these theories
along with a suitable motivation, I would
have perhaps implied myself ...
Too
much courted theories
- N: It is nevertheless a very strong
word. repulsion.
- T: Yes, it is a strong word but, you
know, it is almost a nearly sociological mechanism;
I think to the theory of the groups of DREGS:
BOURBAKI, at that time ,it was only spoken of that,
in the 1955's and every one was very incited
, I definitely, I always had this sort of
feeling that, when a theory is too much adulated, I
prefer not to take charge of it; it is as when a
woman is too beautiful, she has too many suitors ,
Indeed, generally, it seems to me an insuperable
obstacle. There are theories which have been
too much courted and when a theory was too much
courted, I deviated from it...
- N: Why?
- T: Oh! I do not know; maybe because
I had actually the feeling of not being at the
level of the competition, on one hand, and then
maybe also the feeling which we could do as well
somewhere else , in less known .areas
- N: You compare mathematics to a
woman...
- T: Yes, it does not perhaps lack
foundation totally ... There are jagged theories
and bonny theories. Finally it is not maybe
correct, I would say rather that there are clean
theories and dirty theories, and on my own I
always sympathise more with a dirty theory. The
clean theories are the theories where things are
set a, where the concepts are clearly outlined, the
problems are also more or less well defined.
Whereas the dirty theories are theories where one
does not know very well where one goes, one
does not know how to organize things and where are
the main directions etc. From such a point of view
, indeed, I was never Bourbakiste, because BOURBAKI
is fond of clean things; I think that it is
necessary to smear one's hands and sometimes even
more in mathematics.
- N: More?
- T: Yes, well, I mean beyond hands
(laughter).
Be
fore the border
- N: And the "catastrophe" in all that
?
- T: Oh! Good, the disasters are not a
part of mathematics. For me, the theory of the
disasters is not a mathematical theory. If the
theory of catastrophes develops, what is obviously
a postulate, it will give birth to mathematical
theories which will be tools to organise precisely
the models that the theory of the disasters means
to edify.
It is the way I see things, theory of
catastrophe, it is a generator of models aimed
at, as a rule, the most diversified sciences. A
priori, I do not see limitations in the selection
of sciences in a position to adopt models of
catastrophic type; but naturally, these models are
rather sketchy and have an approximate feature to
start with but with the possibility to try to
refine them and elaborate models for which it will
be doubtless needed new mathematical tools; such
new mathematical devices will create probably new
problems.
It is like that I see the theory of the
"catastrophes" as something standing on the
border of mathematics, the border
between mathematics and experimental disciplines,
the disciplines of application.
-N: Actually), it is your place to
stay at the border?
- T: Yes maybe , it is not for nothing
that I made my essential mathematical work on the
notion of edge (laughter), the" bordisme"..,
yes; I presently write a paper which is called "
on the borders of the human power, gambling
".
Edge,
border, limit, peculiarity
- N: From where comes this interest to
edges, borders, centers?
- T: But it is completely natural!: when
you are in a convex structure, you know perfectly
well that your convex is generated by "
extrémal "points. So in many situations,
if you know the situations of "extremal "points ,
you are capable of reconstruing the rest. It is
not only true in mathematics, but even in
completely general situations.
For example, in a socio -cultural environment,
if you look at what newspapers speak about , there
are always "extrémal" situations: the most
sensacional crime, the biggest disaster, etc.
the fascination of the extrémal is something
completely fundamental in the human mind.
- N: But why such a
fascination?
- T: But (laughter)... To reach the
limits of the possible, it is necessary to dream
the impossible, and it is really the
interface between the possible and the
impossible that is important because if we know
it, we know exactly the limits of our
power.
In a dynamic system governed by a potential, as
for example, variety of levels , lines of slope of
a landscape, what is important it is the
border of the pond: to know how the space is
divided in the various ponds between its various
attractors . All the qualitative dynamics is a
question of border.
For that purpose, it is necessary to
characterize points, asymptotic regimes which are
the attractors and then characterise the
borders which separate the ponds from the
various attractors.
I think that these two types of problem( as
would say our literary colleagues), are found a
little in all situations, in all disciplines; there
are the stable asymptotic regimes that is
necessary to characterise and then to study the
approach of the unstable regimes, which
is a problem of border. It is finally a problem
of determinism . A situation is determinist if
the border which separates the ponds from the
various exits is regular enough to be designed; and
if the initial datum with regard to this border can
be localised; the problem is therefore resolute.
But if the border is fluctuating, blurred,
etc. then statistical methods should be utilised
which is much more painful. There is no need to
speak a lot to justify the problems of
borders...
- N: It is not so much the problem to
prove, it is the fact to see the interest that you
bear specially in this same problem almost
everywhere...
- T: Yes, yes, it is exact...
- N: There was something in you which
motivated this interest...
- T: Yes, the borders obviously,
is important in itself... But it is a particular
case of peculiarity, isn't it? I spoke just
before about defects, it is clear that
the defects are not borders, but it is
nevertheless very interesting.
- N: What difference do you make
between defect and peculiarity?
- T: Defect, it is the word which
comes essentially from the crystallography and the
metallurgy ? You have an environment which is
perfectly crystalline, but that, in certain places
presents breaks or fractures or walls, all these
local irregularities, are called defects? The
theory of the defects is a theory which is
mathematically very interesting and, in fact, one
can almost even consider that the theory of the
"cohomology " was originated there... In a certain
sense.
- N: Have you the impression that you
are still interested in the same kind of problems:
defects, limits, edges, borders?
- T: I admit that it is a little bit
difficult for me to go back , let us say, twenty
five years behind. I believe that at that time, I
was really amore strictly mathematician, it is
true; I had to learn mathematics and my first
scientific work, for my first publication concerned
the theory of Morse. And it was also, somewhat,
a correspondence between defects and
peculiarities... and the cellular decomposition of
a space. There was there almost in germ and the
idea too that the study of the peculiarities gives
a means of access to the understanding of a space;
every peculiarity, as a matter of fact, displays in
a space which is appropriate and which it drags
with , in a sense. Then in the case of a
minimum, a sole" attractor," you opened all
trajectories aimed at this "attractor." But for
different peculiarities, for example for the
peculiarity of type collar, there are the dividing,
etc. There is always a sort of configuration
satellite associated to a peculiarity...
- N:... Which characterises almost
...
- T:... Which characterises the
peculiarity, yes. And at this moment, the total
space becomes the meeting point of all the
configurations satellites of these
peculiarities.
An
universe in which there would be an eternal
come-back
- N: And during your schooling was it
under whatever form ,that problems interested
you?
- T: Oh! At that period of time, I was
more school-minded, I think. I do not remember
having thought of things under this form.
But I remember that when I was about seventeen
years old I began to be interested in tdynamics. I
do not remember any longer in which occasion I had
handed a paper to my professor of elementary maths-
in which I spoke of eternal come-back seen
through a dynamic point of view, theories of
eternal return...
It was the idea that one could have a
space-time, an universe in which there would be
eternal return, that is where the dynamics
would be periodic, but I believe that it is about
the first time when I thought really things in term
of dynamics...
Extract from the book: "
Interviews
with
mathematicians
"
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