History of mathematics' didactic
What is the mental representation?
Matematical mental representation
Recording interview: René Thom, Dominique, Rosine...
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Meeting René THOM

Field medal award

( Nobel prize for mathmatics) )

How the motivation of a discipline fits the personality

(Translation by Michel DURAND)


Subtitles and bold types are of my own responsibility.The bold types indicate that what I mean as an emotional aspect, according to my point of view ,therefore belonging to the category of the "cognitive-emotional" interactions

T = René THOM / N = Jacques NIMIER

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 "

Motivation for discipline is not ground for " gadgets ".

It is fixed in the personality of the subject through the representation which he has of this discipline.


Recording interview with teacher:

Dominique and maths/tool

Rosine and her confinement in mathematics

Brigitte and math: the mainstay of life

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History of mathematics' didactic
What is the mental representation?
Matematical mental representation
Recording interview: René Thom, Dominique, Rosine...
Write to me