Abstract
In this chapter, I will discuss René Thom’s approach to philosophy based on his mathematical background. At the same time, I will highlight his connection with Aristotle, his criticism of the modern view of science as a predictive process, his ideas on mathematical education, his position with respect to the French school of mathematics that was dominant in his time, and his relationship with the philosophical community. I will also touch upon the connections between Thom’s ideas and those of Leibniz, Riemann, Freud and others.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
The city of Montbéliard oscillated a few times between France and Germany. From 1042 to 1793, it was part of the Holy Roman Empire, after which it was attached to France. It was occupied by the Prussians during the Franco-Prussian War of 1870, and liberated one year later by the so-called Bourbaki army, named after Charles-Denis Bourbaki (1816–1897), a general of the army of Napoleon III and son of the Greek colonel Constantin-Denis Bourbaki. The latter was educated in France and he served in the French army between 1787 and 1827. The pseudonym Nicolas Bourbaki, which was chosen randomly, refers to the name of general Bourbaki.
- 2.
The translations from the French are mine. For some more subtle sentences, I have included the French original in a footnote.
- 3.
I have transformed “classe de troisième et de seconde” of the French educational system into their equivalent in the American one.
- 4.
In the old educational French system, the “baccalauréat”, that is, the national high school diploma, had two parts. The “baccalauréat 2ème partie” corresponded to the last year in high school.
- 5.
Finishing school at age 16 is not unusual in France for a talented child.
- 6.
“Mon père a tiré le diable par la queue toute sa vie…Mes parents se sont vraiment saignés pour me faire faire des études.”
- 7.
The city of Lyon was occupied later by the Germans, in November 1942.
- 8.
This is again peculiar to the French system, where the preparation of the entrance exams to the so-called grandes Écoles is done in specialized classes provided by some high schools, after the usual secondary school program.
- 9.
The Lycée Saint-Louis, before Thom enrolled there, had among its pupils André Weil, Laurent Schwartz, Gustave Choquet and several other preeminent mathematicians.
- 10.
This is a French diploma allowing, in principle, to become a teacher in a high school, but many students who plan to teach at university also pass this exam.
- 11.
In those times, one could enter academia in France without having a doctorate.
- 12.
Thom entered topology just after a major revival of this field that started in the 1930s. Topologists tried to classify manifolds and were looking for algebraic invariants. The notions of fibre space and fibre bundle became central in algebraic topology around the year 1950, and the topology of manifolds passed important milestones in the space of very few years: In 1956, Milnor proved the existence of exotic differentiable structures on 7-dimensional spheres. In 1961, he disproved the so-called Hauptvermutung der kombinatorischen Topologie (“main conjecture of combinatorial topology”), a conjecture formulated in 1908 par Steinitz et Tietze, asking whether any two triangulations of homeomorphic spaces are isomorphic after subdivision. The higher-dimensional analogue of the Jordan curve theorem (the so-called Jordan–Schoenflies theorem) was obtained in 1959–1960 (works of Mazur, Morse and Brown). In 1961, Smale proved the Poincaré conjecture for dimensions ≥ 7. The same year, Zeeman obtained the corresponding result for dimension 5, and the year later, Stallings for dimension 6. Thom considers that he arrived in topology at the right time. He writes in (Thom 2004): “I think that researchers who started in mathematics after 1960 have had infinitely more merit than than people like me who came along at just the right time.”
- 13.
In those days, the task of the thesis advisor was generally that of a guarantor at the moment of the defense. In general, suggesting the subject of the thesis was not part of his role. At any rate, in Thom’s case, it was not Cartan who proposed the topic of the work. I discussed this at length in the article (Papadopoulos 2018a).
- 14.
This was in the summer of 1951. Cartan had arranged an invitation for Thom to Princeton for the following fall, and the thesis had to be defended before.
- 15.
The sub-title of Thom’s book Stabilité structurelle et morphogénèse is Essai d’une théorie générale des modèles (An attempt for a general theory of models).
- 16.
The institute was founded by Léon Motchane (1900-1990), a Russian-born immigrant who started with small jobs in Switzerland and then Germany and who later became a wealthy businessman in banking and insurances. Motchane liked mathematics and he started studying them relatively late in his life, encouraged by Paul Montel. At age 54, he obtained a doctorate, with Gustave Choquet as supervisor, and he decided to dedicate an important part of his time and fortune to the foundation of a mathematical institute. He managed to obtain funding from several large private companies. The institute was founded in 1958, and Thom was among the first professors hired. The Institute was situated in Paris, in a two-room apartment in the Thiers Foundation, located in a private mansion in the 16th arrondissement, before moving four years later to Bure-sur-Yvette, in the domain called Bois-Marie, which became its permanent location. Today, the French government is the main source of funding for the IHÉS. Motchane represented Grothendieck for his reception of the Fields medal at the Moscow ICM (1966).
- 17.
Thom wrote: “initially, the theory of structural stability seemed to me to be of such scope and generality that with it, I could hope to replace thermodynamics by geometry, to geometrize in a certain sense thermodynamics, eliminating from thermodynamical considerations all the stochastic and measurable aspects, to retain only the corresponding geometric characterization of the attractors. It is certain that the instability phenomena of the attractors that have been discovered since then show that such a hope is false or, at least, that it would be necessary to weaken considerably the notion of structural stability.” (Thom 2004, p. 587) (1989).
- 18.
The mathematical formulation of this notion asserts the existence of a conjugacy between a system and close-enough systems so that in some sense the two systems are similar.
- 19.
I would like to point out the interesting paper (Farmaki and Negrepontis 2021) by Farmaki and Negrepontis titled The paradoxical nature of mathematics, in which the authors argue that the deductive strength in Mathematics is strongly related to its paradoxical nature, and in fact, that it comes, maybe exclusively, from its proximity to the contradictory.
- 20.
- 21.
Etymologically, the word “method” consists of two words: the word is a combination of the words μετά, which means “after”, or, “to follow” and ὁδός, which means a “way”, or a “path”.
- 22.
Here and in the following, while I am giving the relevant passages in Aristotle, I am translating Thom’s interpretation of the passages in Greek.
- 23.
I take this opportunity to point out an extremely interesting interpretation of Zeno’s paradoxes by S. Negrepontis in (2019), based on a thorough interpretation of Plato’s dialogue Parmenides. In short, Negrepontis’ explanation is that Zeno’s aim in his argumentation is to show that the sensible Beings are separate from the true/intelligible Beings.
- 24.
Metaphysics (Aristotle 1928) 1027a20.
- 25.
Ce qui caractérise une fonction, ce qui la distingue des autres, c’est l’ensemble de ses singularités. Il en est des fonctions comme des objets concrets et des êtres vivants.
- 26.
Haefliger, in his article (Haefliger 1988), writes, before presenting the ideas contained in this note and its later developments: “This was not too early, because the administrators at CNRS wondered if it was necessary to continue to support this young mathematician who was so little productive. [Il était temps, car les responsables du CNRS se demandaient s’il fallait continuer à soutenir ce jeune mathématicien si peu productif.]
- 27.
In the 1970s, Heisuke Hironaka built a theory on the desingularization of of sub-analytic sets, showing that locally, any such set is a finite union of images of spheres glued together by real-analytic maps. The theory had been foreseen by Thom. See Hironaka’s exposition in (Hironaka 1973).
- 28.
In the Politics (Aristotle 1908) 1253a9-10, Aristotle considers that what separates man from other animals is the fact that man is a reasonable animal, λόγον ἔχον, “endowed with logos”, the word logos meaning here speech, or reasoned speech.
- 29.
Fragment B93, cf. (Héraclite 1988).
- 30.
Although, as in well known, there is an uncertainly concerning the authorship of the whole treatise, there is not doubt on the fact that it belongs to the Aristotelian school and that it was inspired, if not completely written, by Aristotle himself.
- 31.
Mersenne’s Harmonie universelle is organized into propositions and proofs, like in mathematical treatises.
- 32.
Il n’y a de science que dans la mesure où l’on plonge le réel dans un virtuel contrôlé. Et c’est par l’extension du réel dans un virtuel plus grand que l’on étudie ensuite les contraintes qui définissent la propagation du réel au sein de ce virtuel.
- 33.
Pour atteindre les limites du possible, il faut rêver l’impossible, et c’est réellement l’interface entre le possible et l’impossible qui est important parce que si nous le connaissons, nous connaissons exactement les limites de notre pouvoir.
- 34.
The example is often attributed to Aristotle, but the latter, in a passage of the Poetics, quoting this sentence, refers to Empedocles, who said that old age is the sunset of life.
- 35.
- 36.
A similar agricultural metaphor was made by Plato in the Timaeus (Plato 1937) (77c7-8), about blood irrigation in humans.
- 37.
P. Cassou-Noguès quotes this letter as being part of the Gödel collection held in the library of the Institute of Advanced Study in Princeton. The letter is also quoted by Hao Wang in (Wang 1996, p. 4).
- 38.
Leibniz used several names for the new field he had in mind: analysis situs, geometria situs, characteristica situs, characteristica geometrica, analysis geometrica, speciosa situs, etc.
- 39.
The two men had met for the first time in 1672, in Paris, where Huygens was settled since 1666. Huygens was 17 years older than Leibniz, and for some time he was his private teacher in mathematics.
- 40.
It is fair to add that several years later, Arnold changed his mind concerning Thom. In an interview published in avril 1997 (Liu 1997), he writes: “I am deeply indebted to Thom, whose singularity seminar at the Institut des Hautes Études Scientifiques, which I frequented throughout the year 1965, profoundly changed my mathematical universe. I was always delighted by the way in which Thom discussed mathematics, using sentences obviously having no strict logical meaning at all. While I was never able to completely free myself from the straitjacket of logic, I was forever poisoned by the dream of the irresponsible mathematical speculation with no exact meaning.”
- 41.
The French doctoral system (doctoral d’état) required the defense of two dissertations. The first one was the main thesis, and the subject of the second one was proposed by the jury of the doctorate, a few months before the defense. Its preparation was supposed to take only a few months (usually 3 to 4).
References
Aristote (1993) Problèmes, texte établi et traduit par Pierre Louis, 3 vol. Les Belles Lettres, Paris
Aristotle (1908) The politics (trans. Jowett B). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press
Aristotle (1909) Poetics (trans. Bywater I). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press, Oxford
Aristotle (1910) History of animals (trans. D’Arcy Wentworth Thompson). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press
Aristotle (1924) Rhetorics (trans. Rhys Roberts W). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press, Oxford
Aristotle (1927) On the soul (trans. JA. Smith). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press, Oxford
Aristotle (1928), Metaphysics (trans. Ross, WD). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press, Oxford, 2d
Aristotle (1930) The Physics (trans. Hardie P, Gaye RK). In: Ross WD, Smith JA (eds) The works of Aristotle translated into English. Clarendon Press, Oxford
Arnold VI (1984) Catastrophe theory, translated from the Russian (Original version, 1981). Springer, Berlin/Heidelberg/New York/Tokyo
Atiyah M (2002) special article. In Mathematics in the 20th Century. Bull Lond Math Soc 34:1–15
Atiyah M et Iagolnitzer D (ed.) (1977) Fields medalists’ lectures. World Scientific Series in 20th Century Mathematics. World Scientific et Singapore University Press, Singapore
Atiyah M et al (1994) Responses to: A. Jaffe and F. Quinn, Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics (Bull. Amer. Math. Soc. (N.S.) 29, 1993, No. 1, p. 1–13), Bull. Amer. Math. Soc. (N.S.), 30(2):178–207
Borel É (1939) Traité du calcul des probabilités et de ses applications, t. 4, fasc. 3: Valeur pratique et philosophie des probabilités. Gauthier-Villars, Paris
Cassou-Noguès P (2007) Les démons de Gödel: Logique et folie. Seuil, Paris
Cavailles J (1940) Du collectif au pari. Rev Métaphys Morale 47:139–163
de Risi V (2007) Geometry and Monadology: Leibniz’s Analysis Situs and philosophy of space. Birkhäuser
Echeverrìa J (1979) L’Analyse géométrique de Grassmann et ses rapports avec la caractéristique géométrique de Leibniz. Studia Leibnoziana 11:223–273
Farmaki V, Negrepontis, S (2021) The paradoxical nature of mathematics. In: Papadopoulos (ed) Topology and geometry. A collection of essays dedicated to Vladimir G. Turaev. European Mathematical Society, Berlin, pp 599–642
Farmer Y (2010) Topologie et modélisation chez René Thom: l’exemple d’un conflit de valeurs en éthique. Philosophiques 37(2):369–386
Haefliger A (1988) Un aperçu de l’œuvre de Thom en topologie différentielle (jusqu’en 1957). Publ Math IHES 68:13–18
Hamburger J (ed.), La philosophie des sciences aujourd’hui, Paris, Gauthier Villars, 1986.
Heath AE (1917) The geometrical analysis of Grassmann and its connection with Leibliz’s characteristic. The Monist 27:35–62
Héraclite (1988) Fragments (trans. Weil S), In Dieu dans Héraclite, Œuvres complètes de Simone Weil, éd. A. A. Devaux et F. de Lussy, IV. 2. Gallimard, Paris, pp 144–146
Hestenes D (1996) Grasmann’s vision. In: Schubring G (ed) Hermann Gunther Grasmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Kluwer Academic Publishers, Dordrecht/Boston, pp 191–201
Hironaka H (1973) Introduction aux ensembles sous-analytiques (rédigé par André Hirschowitz et Patrick Le Barz). In Singularités à Cargèse, 1972, Astérisque, Nos. 7-8, Soc. Math. France, Paris, pp 13–20
Huygens Ch. (1899) Œuvres complètes, t. VIII. Correspondance 1676–1684 (ed. Johannes Bosscha jr.). Martinus Nijhoff, Den Haag
Jaffe A, Quinn F (1993) Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics. Bull Am Math Soc 29:1–13
La pensée mathématique, Séance du 4 février 1939, exposés de Jean Cavaillès et Albert Lautman, discussion avec H. Cartan, C. Chabauty, P. Dubreil, C. Ehresmann, M. Fréchet, J. Hyppolite, P. Lévy et P. Schrecker, Bulletin de la Société française de philosophie, 40e année, 1946, pp 1–39
Lautman A (1977) Essai sur l’unité des mathématiques et divers écrits. Union générale d’éditions, Paris
Leibniz GW (1840) Nouveaux essais. Avant-Propos. Opera philosophica 1:198
Leibniz GW (1849) Leibnizens Mathematische Schriften. Edited by C. I. Gerhardt. Verlag von A. Asher & Comp, Berlin
Leibniz GW (1995) La caractéristique géométrique, text edited and annotated by J. Echeverrìa, translation, notes and postface by M. Parmentier, Coll. Mathesis. Vrin, Paris
Lejeune C (1978) Du point de vue du tiers …, in Circé. Cahiers de recherche sur l’imaginaire, 8-9, éditions Lettres Modernes, Paris, pp 91–118
Liu SH (1997) An interview with Vladimir Arnold. Notices of the AMS 44:432–438
Manin YI (1990) Mathematics as metaphor. In: Proceedings of ICM, Kyoto, Mathematical Society of Japan, vol II, pp 1165–1171
Manin YI (2007) Mathematics as metaphor: Selected essays of Yuri I. Manin American Mathematical Society, Manin
Mersenne M (1965) Harmonie universelle, contenant la théorie et la pratique de la musique, 3 vol., Paris, Sébastien Cramoisy, 1636; reprint. CNRS Éditions, Paris
Montel P (1932) Leçons sur les fonctions entières ou méromorphes. Gauthier-Villars, Paris
Negrepontis S, Plato on geometry and the geometers, In Geometry in history, ed. S. G. Dani and A. Papadopoulos, Springer, Cham 2019, p. 1-88.
Nimier J (ed) Entretiens avec des mathématiciens (l’Heuristique mathématique). Publication de l’IREM de Lyon, Villeurbanne
Papadopoulos A (2018a) René Thom, des mathématiques à la biologie et à la philosophie. In: Thom R, Portait mathématique et philosophique. CNRS éditions, pp 19–149
Papadopoulos A (ed) (2018b) René Thom, Portait mathématique et philosophique. CNRS éditions, 460 pages
Papadopoulos A (2018c) Logos et analogie: notes sur deux thèmes récurrents dans la pensée de Thom, In René Thom, Portait mathématique et philosophique (A. Papadopoulos, ed.). CNRS éditions, pp 261–321
Plato (1937) Timaeus translated with commentary by F. M. Cornford. Kegan Paul, London
Porte M (1990) Psychanalyse et sémiophysique: Études épistémologiques en métapsychologie et en dynamique qualitative, thèse de doctorat, Université de Paris 7
Porte M (1994) La dynamique qualitative en psychanalyse, with a Preface by René Thom. Presses Universitaires de France, Paris
Porte M. Notule biographique (notes pour une biographie de Thom), unpublished, available on the internet
Thom R (1949) Sur une partition en cellules associée à une fonction sur une variété. C. R. Acad. Sci. Paris 228:973–975
Thom R (1952) Espaces fibrés en sphères et carrés de Steenrod. Ann. Sci. Éc. Norm. Supér 69:109–182
Thom R (1962) Sur la théorie des enveloppes. J Math Pures et Appliquées, t. XLI, fasc. 2, pp 177–192
Thom R (1970) Les mathématiques modernes: une erreur pédagogique et philosophique? L’Âge de la Science, III, 3, Paris, Dunod, pp 225–242, Modern mathematics: an educational or philosophical error? Amer. Sci., 59, 1971, pp. 695–699. New directions in the philosophy of mathematics, pp. 67–78, Birkhäuser Boston, Boston, MA (1986); Translated from the French in Fiz.-Mat.-Spis.-Blgar.-Akad.-Nauk., 15 (48) (1972), pp. 151–161 (Bulgarian); Translated from the French in JN: Obzornik-Mat.-Fiz., 19, 4, pp. 161–168 (Slovenian). (There are other translations.)
Thom R (1975) Topologie et linguistique. In: Essays in topology and related topics, mémoirs dedicated to Georges de Rham, Springer, 1971, pp 226–248. Russian version, transl. by Yu. Manin, in Uspehi Mat. Nauk 30, n° 1, 181, pp 199–221
Thom R (1972) Stabilité structurelle et morphogénèse: Essai d’une théorie générale des Modèleo, Mathematical physics monograph series. W. A. Benjamin, Inc., Reading. Advanced Book Program, Paris, Ediscience
Thom R (1973) De l’icône au symbole. Esquisse d’une théorie du symbolisme. Colloque de Mons, Le rôle du symbole en mathématiques, 28-29 avril 1972, Belgique, in Cahier intern. du symbolisme, 22/23, pp 85–106
Thom R (1974a) La linguistique, discipline morphologique exemplaire. Critique 322 René, pp 235–245
Thom R (1974b) Modèles mathématiques de la morphogenèse, 1st edn. Christian Bourgois, Paris
Thom R (1975a) La science malgré tout. Encyclopaedia Universalis, Organum
Thom R (1975b) Les mathématiques et l’intelligible. Dialectica 29:71–80
Thom R (1978a) La double dimension de la grammaire universelle, AKUP, Cologne University, 1977, Published also in Circé. Cahiers de recherche sur l’imaginaire, 8-9, éditions Lettres Modernes, Paris, pp 78–90
Thom R (1978b) Les archétypes en biologie et ses avatars modernes, Bulletin de la société ligérienne de philosophie, 1976. Published also in Circé. Cahiers de recherche sur l’imaginaire, 8-9. éditions Lettres Modernes, Paris, pp 52–64
Thom R (1978c) La notion d’archétype en biologie et ses avatars modernes, Bulletin de la société ligérienne de philosophie, 1976. Published also in Circé. Cahiers de recherche sur l’imaginaire, 8-9. éditions Lettres Modernes, Paris, pp 25–39
Thom R (1978d) Le statut épistémologique de la théorie des catastrophes, exposé au séminaire de philosophie et mathématique, ENS, Paris, 21 mars 1976. Circé, Cahiers de recherche sur l’imaginaire, 8-9, Paris, éditions Lettres Modernes, pp 7–24
Thom R (1978e) Les racines biologiques du symbolisme, Conférence à l’Institut de recherche sur l’imaginaire, Chambéry 21-22 mai 1976, Circé, Cahiers de recherche sur l’imaginaire, n° 8-9, Paris, Éditions Lettres Modernes, pp 40–51
Thom R (1980) Paraboles et catastrophes, Entretiens sur les mathématiques, la science et la philosophie réalisés par G. Giorello et S. Morini, Paris, Flammarion, collection Dialogues, 1983. First Italian edition, italienne, Milan, Saggiatore
Thom R (1981) Mathématiques essentielles, École d’automne de biologie théorique, Abbaye de Solignac, 14 septembre - 2 octobre
Thom R (1985) La méthode expérimentale: un mythe des épistémologues (et des savants)?, Comptes Rendus de l’Académie des Sciences, série générale, La vie des sciences, II, 1, Paris, Gauthier-Villars, janv.-fév. pp 59–68
Thom R (1987a) Considérations sur la finalité: d’Aristote à Geoffroy Saint-Hilaire. Hommage au Pr. Pierre-Paul Grassé. Evolution, Histoire, Philosophie, Paris, Masson, pp 155–160. D’après S. Khaznadar, conférence à Osaka, colloque Structuralist Biology, décembre 1986
Thom R (1987b) Mathématique et réalité: faut-il croire Aristote? Colloque de la Société Wittgenstein, Kirchberg am Wechsel, 4-13 août 1986. Publié in Logik, Wissenschaftstheorie und Erkenntnistheorie. Akten des 11. internationalen Wittgenstein Symposium, Wien, Holder-Pichler-Tempsky, 1987, pp 115–121
Thom R (1988a) Philosophie de la singularité. Singularities, Banach Cent. Publ. 20, 7–14
Thom R (1988b) Les intuitions topologiques primordiales de l’aristotélisme, Revue Thomiste, 88, 1988b, n° 3, p. 393-409.
Thom R (1988c) Un rapport problématique: la rencontre théorie-expérience. In The role of experience in science, Actes d’une conférence de l’Académie internationale de philosophie des sciences tenue à Heidelberg en 1986. de Gruyter, Berlin, pp 181–189
Thom R (1989a) Causality and finality in theoretical biology: a possible picture. Proceedings of the conference at Abisko, Sweden, 4-8 May 1987. In Newton to Aristotle. Toward a theory of models in living systems, J. Casti et A. Karlqvist ed. Springer, New York, pp 39–45
Thom R (1989b) Problèmes rencontrés dans mon parcours mathématique: un bilan. Publications mathématiques de l’IHÉS 70:199–214
Thom R (1976) The twofold way of catastrophe theory, In Structural stability, the theory of catastrophes and applications in the sciences, Springer Verlag, Lecture notes in mathematics, no. 525, pp 235–252. French transl. Les deux voies de la théorie des catastrophes, In Apologie du Logos, Paris, Hachette, 1990a, pp 373–394
Thom R (1990b) Esquisse d’une sémiophysique: Physique aristotélicienne et théorie des catastrophes, Paris, InterEditions, 1988. English translation by Meyer V, Semio Physics: A Sketch. Aristotelian Physics and Catastrophe Theory, Addison-Wesley
Thom R (1990c) Homéomères et anhoméomères en théorie biologique d’Aristote à aujourd’hui. Conférence à l’île d’Oléron, Biologie aristotélicienne, 26–30 juin 1987. Biologie, Logique et Métaphysique chez Aristote, éd. P. Pellegrin, Éditions du CNRS, Paris, pp 491–511
Thom R (1990d) Les réels et le calcul différentiel, ou la mathématique essentielle, Talk given at the Séminaire de philosophie et mathématiques, École Normale Supérieure, 17 mars 1982, in Apologie du Logos, Paris, Hachette, 1990d, pp 314–331
Thom R (1988) Structure et fonction en biologie aristotélicienne, in Biologie théorique, Solignac, Paris, éditions du CNRS, 1993, also published in Apologie du logos, Paris, Hachette, 1990e, pp 247–266
Thom R (1991a) Matière, forme et catastrophes. In: Sinaceur MA (ed.) Penser avec Aristote. érès, Paris, pp 367–398
R. Thom (1991b) Saillance et Prégnance, In: L’inconscient et la Science, ed. R. Dorey, Dunod Paris, 64–82
Thom R (1991c) Prédire n’est pas expliquer. Entretiens avec Émile Noël, Paris, Eshel, coll. La Question
Thom R (1992a) La théorie des catastrophes, interview with Francis Bouchet. INA, Eshel, Paris
Thom R (1992b) Actualité de la physique aristotelicienne. In: Geometry and analysis in nonlinear dynamics Proc. Workshop Chaotic Dyn. Bifurcations, Groningen/Neth. 1989, Pitman Res. Notes Math. Ser. 222, 85–96
Thom R (1992c) Leaving mathematics for philosophy. In: Mathematical research today and tomorrow (Barcelona, 1991), 1-12, Lecture notes in Math., n° 1525. Springer, Berlin
Thom R (1992d) La théorie des catastrophes, entretien avec Francis Bouchet. INA, Eshel, Paris
Thom R (1994) The task of Mathematics in describing the world, Proceedings of the symposium 25 Años de Matematicas en la Universidad de la Lagune, Secretariado de Publicaciones de la Universidade de La Laguna, Tenerife, 21–25 novembre, pp 83–86
Thom R (1995) Biologie Aristotélicienne et topologie. Boletim da Sociedade Portuguesa de Matemática 32:3–14
Thom R (1996a) Histoire de la transversalité, d’Aristote à la théorie des catastrophes. In: Universalité de la transversalité, a talk given at the Journée de réflexion du 6 février: La transversalité dans la Géométrie Moderne, Maison des Sciences de l’Homme, Paris
Thom R (1996b) Qualitative and quantitative in Evolutionary Theory with some thoughts on Aristotelian Biology. In Systematic Biology as an Historical Science. Memorie della Società Italiana di Scienze Naturali e del Museo Civico di Storia Naturale di Milano XXVII(I):115–117
Thom R (1997) The hylemorphic schema in mathematics, In Philosophy of mathematics today, Episteme 22. Kluwer Academic Publishers, Dordrecht, pp 101–113
Thom R (1998) Comment la biologie moderne redécouvre la kinèsis d’Aristote. Bull Hist Epistém Sc Vie 4(3):197–208
Thom R (1999) Aristote topologue. Rev synth 4e s., n° 1, pp 39–47
Thom R (1989) Exposé introductif. In: Logos et théorie des catastrophes, Petitot J (ed.), Actes du colloque international de Cerisy de 1982, Centre culturel international de Cerisy-la-Salle, Genève, Patiño. Published also in Gazette des mathématiciens, special edition, 2004, pp 7–22
Thom R (2017) Œuvres mathématiques, vol I. Société mathématique de France
Thom R (2021) Métaphysique extrême; entretiens. In: Onicar? Foucault, Duby, Dumézil, Changeux, Thom: Cinq grands entretiens au champ freudien, Navarin éd., pp 51–101
Thom R, Lejeune C, Duport J-P (1978) Morphogenèse et imaginaire. Lettres modernes, Circé, Coll. Méthodologie de l’imaginaire, Paris
Wang H (1996) A logical journey: from Gödel to philosophy. The MIT Press, Cambridge, MA
Zeeman EC (1965) Topology of the brain. In: Mathematics and computer science in biology and medicine. H.M. Stationary Office, London
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Additional information
Toute ma métaphysique sous-jacente, c’est d’essayer de transformer le conceptuel en géométrique, le logique en dynamique.
(René Thom, from a letter to Claire Lejeune, February 20 1980, quoted in (Lejeune 1978))
Appendix: Jean Cavaillès and Albert Lautman
Appendix: Jean Cavaillès and Albert Lautman
Let me say a few words on the two philosophers of science, Jean Cavaillès (1903–1944) and Albert Lautman (1908–1944) to whom Thom was attracted as a young student.
Cavaillès is the author of an important philosophical corpus on the foundations of mathematics in which the stress is on a dynamical evolution of these foundations. Among the multitude of philosophical schools of the first half of the twentieth century, Cavaillès was situated between Hilbert’s formalism and Brouwer’s intuitivism, a trend that he used to call “modified formalism”. He was close to the mathematician Émile Borel, one of the French representatives of the “semi-intuitionist” current. As a matter of fact, in 1940, Cavaillès wrote an authoritative synthesis of Borel’s theory of the quantification of chance (Cavailles 1940), which constitutes the latter’s point of view on the philosophy of probability theory expressed in Vol. IV of his Traité du calcul des probabilités et de ses applications (Borel 1939). The volume is titled Valeur pratique et philosophie des probabilités.
Cavaillès taught at the École Normale Supérieure before Thom entered there. His philosophical work, which was in part the result of a close collaboration with Emmy Neother, had a non-negligible impact on the mathematical research done at the École.
Lautman had been Cavaillès’s student at the École Normale. In 1937, he defended a doctoral dissertation, in two parts, the first one titled Essai sur les notions de structure et d’existence en mathématiques (Essay on the notions of structure and existence in mathematics), and the second one, Essai sur l’unité des sciences mathématiques dans leur développement actuel (Essay on the unity of the mathematical sciences in their present development).Footnote 41 Following the path of Cavaillès (and before him, that of Poincaré), Lautman considered that both the formalist and the intuitionist movements were a failure. He was an advocate of structuralism in the tradition of Bourbaki. He was a promoter of the concept of unity of mathematics, see the collection of articles (Lautman 1977).
Cavailles and Lautman found the sources of their theories in the recent developments in mathematics and physics (notably quantum physics). They both wondered about the role and the consequences of the various movements of thought that had appeared at the end of the nineteenth century on the philosophy of mathematics (conventionalism, logicism, constructivism, formalism, etc.). Even if they diverged on some points, and in particular on the organization of mathematics as a system of thought, they both considered that the philosophy of mathematics must necessarily be at the center of any metaphysical theory, as it was already for Plato, Heidegger and other philosophers. Their work is an embodiment of this approach. Both Cavailles and Lautman had close relations with mathematicians of the Bourbaki group such as Cartan, Chevalley, Dieudonné, Ehresmann, and Weil, and they were very much interested in Boubaki’s project of writing complete treatises on the foundations of several topics in mathematics.
During the Second World War, Lautman et Cavaillès joined the resistance to the German occupation of France. They were both shot by the Nazis in 1944. The first was 36 years old and the second 40.
The French Society of Philosophy devoted its session of February 4, 1939 to the discussion of Cavaillès and Lautman’s works. Several mathematicians, including Henri Cartan, Paul Dubreuil, Paul Lévy, Maurice Fréchet, Charles Ehresmann, and Claude Chabauty, were present at that session. A report on this session was published after the liberation of France, in the Bulletin de la Société française de philosophie (La pensée mathématique 1946) (1946). Thom may have been present at this debate, but being still too young, his name is not mentioned in the Annals.
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this entry
Cite this entry
Papadopoulos, A. (2022). René Thom: From Mathematics to Philosophy. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_98-1
Download citation
DOI: https://doi.org/10.1007/978-3-030-19071-2_98-1
Received:
Accepted:
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19071-2
Online ISBN: 978-3-030-19071-2
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering