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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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