The Body of René Thom (singularities)

Reading time: 22 minutes

Translation by AB – March 21, 2021


René Thom (1923-2002), a French mathematician who won the Fields Medal in 1958, sailed far beyond the mathematical continent. From questioning to questioning, after having come across biology, sociology or linguistics, Thom ended up by discerning a “system of the world” and it is as a philosopher that he ended his journey. Somewhat forgotten, his heterodox vision now deserves the attention of those who still question the technical claim of the “digital” to fully underpin our environment and our lives.

We must see, for example, how René Thom engages his own body in his mathematical work, a body that makes him an absolutely “non-digitizable” practitioner (Ecce Homo Mathematicus). We have to follow him in order to come into contact with “singularities”, those mathematical objects which can only be invented and grasped in a kinesthetic way. Any natural phenomenon can only appear bordered by these singularities, which are like ruptures, edges, accidents… It is “where” for us, living sentient organisms, something happens. Now, for mathematical reasons, Mundus Numericus cannot produce any authentic singularity, nor therefore any authentic event. Our digital world is a phantasmagoria that even the term “artificial” cannot account for.

We are going to explore all of this a bit by following in the footsteps of this singular mathematician, but without ever doing mathematics.


René Thom gained some notoriety when a (relatively) large audience discovered his famous “Catastrophe theory” developed in the second half of the 1960s. This theory seemed accessible with its simple formulas and its diagrams with poetic names (fold, cusp, swallowtail…). Its universal language, descriptive and qualitative, made it possible to establish unexpected dialogues between scientific districts that had been autarkic until then: physics, linguistics, biology, psychoanalysis… This little revolution instigated from mathematics agitated for some time the cultural circles and even inspired, for the anecdote, artists like Salvador Dalí (the “Series on Catastrophes”, ten paintings devoted to the works of René Thom) or the film-maker Jean-Luc Godard (“René”, medium-length film realized in 1976)…

However, this theory has not seen any real posterity. What might be better called a “theory of models” (or “shapes”) offers “only” one way of describing and explaining what is going on. It is not capable of any prediction, does not participate in any artifact, has no performative power. Although mathematical, it is as indifferent to the technical system as if it was a philosophical production. In the words of Wolfgang Wildgen, a professor of linguistics at the University of Bremen, René Thom was aiming for “the transition to “soft” theorizations, therefore a qualitative analysis, but exact from a mathematical point of view, a validation centered on explanation, understanding (“on a human scale”)1. But what one might call an “ethics of clarity” weighs little in the face of the technical imperative of effectiveness (although we are “begging” today, finally, for algorithms and other “artificial intelligences” to explain their decisions!).

René Thom has thus produced a qualitative mathematical material that can be applied to a multitude of phenomena such as the breaking of a wave, predation, embryogenesis or even the meaning of words… There are indeed universal shapes that only mathematics can reveal, and which would explain underlying coincidences between fields that modern science has separated. At the heart of the dynamics of these universal shapes, we are going to meet these famous singularities, these “places” where something happens. But let us begin with a few words about René Thom himself.


René Thom was born on September 2, 1923 in Montbéliard (eastern France) where he spent his childhood and adolescence. His family lived in a “large building with a spiral staircase that was called a “yorbe” in the Montbéliard dialect2. There, a certain Uncle Georges seems to have had a great impact on René Thom. According to the notes taken by the psychoanalyst Michèle Porte during her interviews with the mathematician, this eccentric uncle was a spiritualist and “was considered an idiot in the family”. And “he recited all his formulas, but the liberation of the body, the liberation of the body from the soul, it takes time!”. René Thom concludes about this uncle: “It gives spirit of freedom”, a spirit of freedom that he has indeed largely inherited.

René Thom began his schooling at the elementary school of Montbéliard in 1931, where he already began to show extraordinary intellectual qualities: “I believe that I had arrived at a very good intuition at that time, and I could already see in four-dimensional space at the age of ten, eleven years3. This is quite exceptional when we know that most of us will never manage to do so4. Space will be his major concern and he will develop a particular aptitude for geometry as early as the 9th grade (emphasis added):

My teacher was not a particularly brilliant man, but he had succeeded in arousing my interest and I really enjoyed it, I was doing very complicated problems of construction of triangles, etc. and it is a bit, essentially, out of nostalgia for that time that I defend elementary geometry against the modernists5.

This “confession” sheds some light on René Thom’s intellectual dynamics. “Nostalgia” is undoubtedly the permanent trace of the pleasure, discovered in childhood, of travelling through space in imagination. This pleasure is not a simple superficial and fortuitous joy but rather a source and a driving force. The “nostalgia for pleasure” morphed into conviction and fueled the effort. As for the “modernists”, they are the great inventors of abstract concepts and symbols who, far from the pure geometrical sensibility, have taken the mathematical continent by storm since the beginning of the 20th century: algebraists, logicians… But Thom aims more generally at all areas of use of abstract concepts because according to him “the ultimate foundation of scientific objectivity is the space-time reference6. An unshaped concept escapes any geometrical censorship (it is “trans-spatial”) and then serves ideology more than science. René Thom even compares formal thinking, a pure language game, to “magical thinking” which prototype is action at a distance (Uncle George is not far away!). In particular, he cites as an example biology “which language is full of words such as: order, disorder, complexity, information, code, message… all these concepts [ having ] the common character of defining long-range spatio-temporal relations“. The shaping of these concepts, in the literal sense of giving them a mathematical shape, as well as their spatio-temporal anchoring are the essential conditions of their scientific objectivity.

According to René Thom, mathematicians would therefore be invested with a civilizing mission accomplished in the shadows7:

The historical merit of mathematicians is perhaps simply that they have progressively brought out those fundamental concepts which are implicit in the mind and which only come to be made explicit as a result of successive historical shocks. The curious thing is that, in the history of science, the physicist is always given much more credit than the mathematician. I am sorry for this as a mathematician, but it is undeniable. The thing is what it is. Everybody has heard of Einstein; few people have heard of Riemann. It is a sociological fact. There is nothing we can do about it.

René Thom thus sees himself as “releasing fundamental concepts” and explaining them in his Catastrophe theory. In this theory, singularities, mathematical objects already well known in his time, play a great role. We will now try to approach them without making any mathematical noise.


It is thus because phenomena in the broad sense (biological, physical, linguistic…) manifest themselves on the stage of a real and continuous space-time that they cannot be correlated at a distance, trans-spatially. “Time” must be taken to traverse the “space” that constitutes them. But above all, if these phenomena appear to us, it is because they occur in the form of a rupture, of a singularity, which detaches itself from the continuous background and activates the senses. We can figure a general singularity in the following way8:


Any system manifests its transition from one state to another by the appearance of a singularity on the phenomenological stage. A singularity does not occupy an infinitely punctual “place” but possesses, so to speak, enough “thickness” to locally compress the global dynamics from which it results. It also has enough space for an “internal stir”, imaginative source of new deployments. A singularity is thus always at the same time a reality and a virtuality. René Thom expresses this idea more rigorously9:

I believe that this is an essential idea, underlying the Catastrophe theory, this Aristotelian idea of the passage from potency to act, which is realized mathematically in this algorithm of deployment of a degenerate germ into a family of maximal germs: the degenerate germ is in a way a virtual structure and when you constitute its deployment, its universal deformation, you obtain a panorama of all the possible current realizations of its virtuality… [ This transformation ] allows a qualitative description of the passage from the virtual to the actual. This is the essential philosophical point brought by the theory of elementary catastrophes and I believe that there will always remain something of it.

Singularities are thus universal phenomena that can be found everywhere. We propose here three examples, in three very different domains, which we believe allow us to understand roughly what it is all about: the experience of the “self”, mathematics and language with, respectively, the singularities that we note ego, point and concept. We will take a little independence with the thought of René Thom’s but his spirit remains, we think, intact.

Ego – A train journey

But, as we shall see, the pleasure of rail travel requires not being in a hurry.

René Thom – Voyage en train et paysage

The “geometrical feeling” can only develop from the experiences of one’s own body and René Thom was singularly inspired by his train journeys. We rely here on a communication made during the colloquium “Arts and Railways”, 3rd colloquium of the AHICF held in November 199310.

The territory over which the train travels is comparable to a surface with declivities, gorges, ridges… and the railway track is a kind of regular mathematical curve on this surface. From then on, the attentive traveler physically experiences this curve by sensations of acceleration (increase or decrease of the train speed but also vertical acceleration when the ground goes up or down, lateral acceleration in the curves…). In his talk, René Thom delivers some interesting comparisons between certain mathematical concepts of topology (an “erudite” extension of geometry) and the surveys of the tracing and the railway device carried out from this point of observation in movement that is his own body, which he calls “ego” and which we note ego (emphasis added)11 :

The movement of the train realizes a kind of gradual expansion of the self, following the cascade of nested referentials:

Ego → compartment → wagon → convay → railway → environment

But the pleasure, if it comes essentially from this dilation of the self, changes of nature and intensity according to the stage where the identification stops. Hence a natural gradation of the factors of pleasure and this change of reference is itself a pleasure.

Pleasure”, again, becomes more precise. It is not only associated to the travelling of his youth through space-time but it is extended to the “dilation” of ego in a deployment of identifications:

Thom thus notes the progressive effects of the dilation of the ego singularity at successive scales that finally lead it to the curve/track and the surface/landscape. This railway example illustrates well the way in which each of us perceives and puts into play his or her own ego, this singularity which, at each moment, manifests itself both as the result of a history and as the promise of immediate possibilities. This deployment can be associated to the pleasure but also, as René Thom will indicate it himself, to the pain. The emotion seems in any case, as far as humans are concerned, to accompany the proper “vibration” of ego.

Ego – Predation

Behind this, René Thom’s mathematical work emerges, without which we would only have to deal with a discourse, an ideology. Between the past of an individual and the perpetual acts of consciousness by which he initiates his deployments, there is no obscure and magic “algorithm” but there is ego, this mathematical singularity, “residual image of the proper body of the observer”, which locates and structures the continuous passage between the past and the future. In his cardinal work “Structural Stability and Morphogenesis” published in 1977, René Thom extends this dialogue, typical of the universal language of catastrophes, between mathematics and biology (emphasis added)12 :

[…] It would be wrong to think that all traces of biological origin have disappeared from mathematical thought. It is striking that most of our spaces, even in pure mathematics, have a point of origin, a residual image of the observer’s own body, in a state of ongoing predation on the environment. Even the typical configuration of Cartesian axes irresistibly evokes a jaw that closes on the prey […]

Here is, perhaps, one of the essential dynamic states of the ego singularity, of biological origin, which causes this dilation of the self of which we find the formal trace in mathematics. Thus, the railway traveler, “point of origin”, is in “state of ongoing predation” and thus dilates to seize virtually the “prey” that is the observed environment: the convoy, the railway, the environment… Note that this seizure is not immediate (i.e., magical or trans-spatial) and that the dilation/expansion of ego and the positioning of its “jaws” requires time.

Ego – Dislocation

These thoughts call for this central theme of Puissance & Raison, which concerns the erosion of the “self” by the multiplication of technical hybridizations. The neurobiologist Francisco Varela and the philosopher Jean-Luc Nancy, both of whom have undergone heavy transplants and hyper-technical medical treatments, have been meticulous observers of this phenomenon (The Mirrors of the “I”). We recalled this word from Jean-Luc Nancy which resonates quite strikingly with Thomian thought (we underline)13:

Between me and me, there has always been space-time; but now there’s the opening of an incision, and the irreconcilable of an upset immunity.

The graft, this foreign body, this intruder, which is now interposed “between me and me”, seems to tear the continuous space-time within which ego appears. Francisco Varela had also experienced these multiple technical incisions by which his “I” seemed to dissociate into so many images, singular points trans-spatially disjointed. In the environment of the delicate Thomian “gradual expansion of the self” the tearing of an ego singularity subjected to multiple simultaneous “journeys” begins:

René Thom also asserts that “all our acts of elementary consciousness are always more or less displacements”, but haven’t these displacements become quite short and more like a kind of “jumps”? Jumps from one screen to another, from one information to another, from an order to a delivery from which the intermediate space-time has vanished, and more generally from one technical means to another… We can suggest that technologies, especially digital ones, expose us to an experience similar to that of Francisco Varela, although more complete: an actual multiplication of technically specialized egos, like so many avatars. This is what some people today call a “dislocating identity”:

Have we for all that exchanged the Thomian “pleasure” of gradual deployment for a kind of pleasure of dislocation? Nothing is less sure. Each ego remains, of course, a “predatory” singularity, conscious of its own potency, and can take pleasure in the scope and harmony of the expansions seen in consciousness. But in the dislocation, each ego also sees this scope, this content of consciousness, reduced, and thus the “quantity” of pleasure associated without the pleasures of each ego being able to add up. It is as if the “surface” on the deployment / dislocation plane was by nature limited, perhaps for biological reasons:

Point – Shaking

The enchantment of the virtual is an enchantment that bypasses magic. […] If the universe cannot be mathematized, it can only be magical or theological. There is not much choice. We never say that, but that is what is important. Here something has shifted, which is gigantic.

Gilles Châtelet

In a little text entitled “Enchantment of the Virtual14, which begins with the words “I’m going to tell you a little about the virtual…”, the French philosopher Gilles Châtelet (1944-1999) gives a wonderful analysis of mathematical singularities, and in particular of the most elementary of them: the point. But before coming to this point-singularity, let’s say a word about this “historical” point that Euclid was already talking about and that we all know. When we give ourselves such a point, when we draw it on a sheet of paper and say “here is a point”, what happens? Nothing at all! And Gilles Châtelet adds:

The point that I choose there, it is me arbitrarily who chose it. It is nothing but me. I am the one who decides of its existence, so I could stay for centuries in front of this point, it would not matter.

The mathematics teacher kept giving us things he had arbitrarily chosen: points, lines, triangles, numbers… But we hadn’t asked him for anything and these objects that came out of nowhere were terribly boring (Châtelet still says: “I define 1, I put it down, it’s over, I just have to die!”). By the way, we were disgusted with this mathematical “Easter Island” with these moai simply put there and devoid of meaning. The given mathematical point is nothing in itself (“it is nothing but by me”). It is inert. Let us recall this definition of Euclid proposed in Body and Language Games by using this font chosen for language games:

A point is anything that has no part.

This “is” is not ontological and this point is only a card in a language game. Something happens only when the body takes hold of it and “shakes it”, so to speak, to reveal its internal power. This point is the point-singularity. Ego can “shake” in unison and consider mimetically the possible expansions of this point full of virtuality. Thus, the point is like a location for potentialities, a trembling source of expanding power, of “fulgurations” says Châtelet following Leibniz (like this: point → and ego →). The essential aspect is that one can choose to see the point as the source of this or that, e.g.:

The points are powers of explosion of lines.

But we could just as easily see exploding circles, squares or laces …


Thus, the points, like all singularities, are “sources of things”, “creators of possibilities”. Châtelet also speaks of this “way that mathematics has, in a completely experimental way, of scratching certain points [ and ] it is really necessary to understand how objects are born from this itch”. It is so well seen, even if it is not the mathematics that scratches but rather the ego singularity in “a state of ongoing predation on the environment”.

Now let’s move on to the last example.

Concept – Magic

A concept in the consciousness, that is to say when we think about it, is also presented as a singularity, that we note therefore concept. It is thus, mathematically, the vibrant point of passage from a global state to another global state:

We think of a cat, or of the truth… and immediately appears the “ahead” of these concepts, their deployments as mental evocations. These deployments depend on each person since a concept in the consciousness condenses a part of the instantaneous global state of the mental space, a space that includes, among other things, the memory and the acquired linguistic competence. This concept corresponds to the local “compression”, on the stage of the consciousness, of a global concept, considered by René Thom as a dynamic structure immersed in a linguistic “substratum” space. Once again, biology is summoned (additions in brackets)15 :

[…] We should not believe that the stability of the meaning is due to the invariance of an inert form, like a printed symbol – a point of view to which the whole formalist philosophy would like to reduce us [ « modernists » ]. On the contrary, it is necessary to conceive that every concept is like a living being that defends its organism (the space it occupies) against the aggressions of the environment, that is to say, in fact, the expansionism of the neighboring concepts that limit it in the substratum space: it is necessary to look at every concept as an amoeboid being, that reacts to the external stimuli by emitting pseudopodia and by phagocytosing its enemies.

It is when this concept finds the “path that brings it back to the speaking subject” that it manifests itself to him as a singularity, a concept, and thus as a power of evocation. According to René Thom, mathematics underlies this dynamic and consequently all linguistic phenomena.

The algorithms manipulate concepts that are not mathematical singularities but the cards of an obscure language game that can therefore be noted as concepts. As the teacher gives inert points for us to do our exercises, these concepts are given to algorithms that chain together the calculations of their language games. To paraphrase Gilles Châtelet, these concepts are “nothing but by” the technological system. There is no transportation of meaningful shapes to us. If a screen or a connected speaker says cat or truth, of course we think of cat or truth and their evocative powers. But the algorithm that has “spoken” ignores mathematical space-time and therefore, freed from any formal constraint, can articulate any magic formula about cats or truth

Is René Thom obsolete?

Again, as Aristotle said, it is not nature that imitates art, it is art that imitates nature. It is because we have implicitly the pattern of the pump made in the heart that we were later able to build technological pumps. And now people tell you, the brain is a computer! We continue…

René Thom16

René Thom was undeniably imbued with a feeling of nostalgia that seemed to push him against the technical and scientific “progress” of his time, with, as the mathematician Marc Chaperon tells us, “a pronounced taste for provocation17. But wouldn’t he be in spite of that the inspirer of a future still unnoticed, of an opening by which we could escape the totalitarian digitized “trans-spatiality” of the humanity, then to the trans-humanism?

There is at least this microscopic opening that we have just glimpsed: a mathematical singularity is not “digitizable”. Indeed, this geometrical, dynamic and metastable being, vibrant source of potentialities where the dialectic of the global and the local, the continuous and the discrete, the virtual and the real is played out, belongs to the continuous space-time (Miguel Benasayag and the question of the “living”). Why does Mundus Numericus still manage to make us believe that something is going on? That it is full of meanings? Apart from ingenuity and technical power, we had foreseen at least two hypotheses.

The first would be the return of an appetite for magical thinking in this “general atmosphere” conducive to the obsessive-compulsive disorder of prognosis and response, an effect of uncertain times, perhaps (Return to Babylon). But the technology that supports Mundus Numericus is only a gigantic language game and therefore massively and instantaneously produces answers (“outputs” we say for an algorithm: things must come out!). They are only answers, not answers, but our compulsion is supplied19.

The second hypothesis is that of a return to the “Bicameral Mind”, to a kind of atrophy of consciousness, this stage on which singularities vibrate (Following Julian Jaynes: comeback of the Bicameral Mind?). Let us recall in two words what it is about. The American psychologist Julian Jaynes (1920-1997) suggested that, a few millennia ago, the right brain “hallucinated” divine voices that commanded the left brain. These voices were suitable for a hyper-stable world, but the environment started to change radically in a few centuries and a more agile and self-aware “I” gradually replaced these voices: consciousness replaced the divine. But maybe the right brain is now taken over by digital “voices”, suitable for a hyper-static Mundus Numericus. Perhaps we are mutating… On this subject, René Thom rightly evoked some works in relation to the lateralization of skills concerning the discrete and the continuous (additions in square brackets)20:

[…] Stephen Kosslyn has studied the case of certain prepositions and has shown that they are processed in two quite different ways, one “continuous” and metrical, the other “categorical” and binary, and that this results in the brain from hemispherical lateralization, continuous treatment preferably taking place in the right hemisphere and categorical [discrete] treatment in the left hemisphere.

The hypothesis of a withdrawal of consciousness, concomitant with the dislocation of ego, could well be supported by that of the taking over of the geometric right brain by digital voices. Would phenomenal singularities of the René Thom’s world, replaced by the instructions of a language game, be only useless vestiges? We could of course be mistaken, for isn’t our “I” always integrated and powerfully deployable? Is our need to travel through space and time not intact? Do we really obey hypothetical “digital voices”? Don’t we always take the time to deploy reasoning against “magical thinking”? … and above all, doesn’t “progress” go in that “right” direction, which is not Thomian?

René Thom was a “shining” mathematician, in the words of Marc Chaperon, and endowed with that “farmer’s common sense”, as we say in France, “that kept him as close as possible to the essential21. His railway meditations led him to regret what he called “modern transportation”, the high-speed train or the plane… But if today we also understand “modern transportation” as digital “trans-portation”, i.e. as “travelling” from one digital point to another via our algorithmic interfaces, then, for us, these last so “obsolete” words take all their meaning22:

Gone are the days of the tram from our childhood which crossed the villages, taking the streets and skimming the houses … We then had the feeling of a trip that we could have done on foot. Modern transportation thinks of us as a package that loses all contact with the outside world on departure and only finds it on arrival. Only the imagination of the continuous path from origin to end, restoring the temporal continuity of the self through space, can bring us back to true life.

1. Wolfgang Wildgen / Estudios Semióticos vol 16, n°3 – December 2020 – L’apport de l’épistémologie de René Thom à la sémiotique
2. For the elements that follow, we have gleaned some information from this Notule biographique written by Michèle Porte, a psychoanalyst and university professor, for a biography of René Thom (which never saw the light of day).
3. Jacques Nimier – 1989 – Entretien avec le professeur René Thom
4. His imaginary space, already richer and more unusual than that of his fellow creatures, was much “airier” than ours, which seems to be shrunken by the digital world: in Mundus Numericus, space is indeed mostly one-dimensional – emails, wires, tweets… -, sometimes two-dimensional – images, maps, smartphone screens… – and occasionally of “2.5” dimensional, if one admits that time passes a little when watching videos. We then have more difficulty with “extra” dimensions that require an additional “motion” effort: shopping, waiting, travelling slowly… – to the point that we may wonder if modern progress is not correlated with a need to reduce dimensional effort…
5. René Thom continues as follows: “As for me, I think that if we persist in the present way, we will deprive ourselves of a method of selection which was really excellent and I would not be surprised if in the years to come, we see a certain drop in the level of mathematics in France following the abandonment of Euclidean geometry”. A prediction that turned out to be quite accurate.
6. Jean Petitot / Mathématiques et sciences humaines, tome 59, p.3-81 – 1977 – Interview de René Thom
7. Annales de la Fondation Louis de Broglie, Volume 27 n° 4 – 2002 – Exposé de René Thom au Colloque International de Cerisy : « Logos et théorie des Catastrophes »
8. All the temporal words – “before”, “after”, “duration” … – and spatial ones – “place”, “before” … – should not be taken literally but as indications for a mathematical “space-time” a little more abstract than ours.
9. Ibid. 7 – Let us add that a mathematical “germ” is not a “singularity”. We assume a confusion which seems acceptable to us within the framework of this article.
10. Karen Bowie and René Thom / Revue d’histoire des chemins de fer, 39 p.293-310 – 2008 – Voyage en train et paysage
11. Ibid. 10
12. René Thom / InterÉditions Paris – 1972 ; 1977 – Stabilité structurelle et morphogenèse
13. Jean-Luc Nancy / Galilée – 2000 – L’intrus
14. Text based on the presentation of June 3, 1986 at the Collège International de Philosophie – Gilles Châtelet / Chimères. Revue des schizoanalyses N°2, été 1987. pp. 1-20; – 1987 – L’enchantement du virtuel
15. Michèle Porte – Citations de René Thom
16. Ibid. 15
17. Marc Chaperon – 2006 – Catastrophes, un témoignage
18. A note for later. The Living (or the “Natural”) could be characterized by a capacity to store “memories” not as recordings but in the form of singularities (or germs), of powers of evocation each time reactivated on the scene of the consciousness. Each reactivation being parameterized by a different “before”, the memory is presented each time in a different way. This is much less costly in terms of storage, but it requires a formal capacity for evocation, perhaps residing, precisely, in the body. A memory is linked to a body and cannot be reactivated without it.
19. Let’s add in passing and for the record that individuals are put into a trans-spatial relationship by the intermediary of algorithms – think of social networks – and that this relationship can therefore only be “naturalized” a posteriori – often by the law.
20. Jean Petitot / – Les premiers articles de René Thom sur la morphogenèse et la linguistique
21. Ibid. 17
22. Ibid. 10

1 Response

  1. 6 November 2023

    […] a main difference: internet is like a juxtaposition of artificial organs (that dislocate us – see The Body of René Thom (singularities)); The Metaverse claims to make up our dislocated “I’s” in a physical and […]

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