Editors: Gabriella Crocco, Frédéric Jaëck
This special issue in two volumes is devoted to a certain French tradition in the philosophy of mathematics, a tradition characterized by the tutelary presence of two major figures, Jean Cavaillès (1903–1944) and Albert Lautman (1908–1944), both of whom were shot by the Nazi occupiers for their involvement in the French resistance, and both of whom produced original, albeit unfinished, works in the course of their short lives.
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Presentation of the second volume dedicated to ‘La philosophie mathématique’: Mathematical and Philosophical Inspirations from Brunschvicg to Granger, a special issue devoted to a certain French tradition in the philosophy of mathematics.
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Our aim in this article is to take the full measure of the formula cited as an epigraph and, through it, to reflect on the problem of the relation between philosophy and mathematics in the work of Jean Cavaillès. In particular, the question will be how to interpret the phrase “by means of mathematics” that appears in this formula. Does this mean that mathematics properly constitutes, for Cavaillès, the framework of expression of thought, or, as we will argue here, that it provides a privileged expression of thought—more “speaking,” though not exclusive?
To approach this problem, we bring to light a double danger: either the philosophical is effaced within the mathematical, or, conversely, the mathematical is subjected to external philosophical norms. The difficulty then consists in situating a theory of science that belongs to the scientific movement while remaining distinct from it. Cavaillès proposes to understand such a theory as an “auto-illumination” of the scientific movement, through which it reveals its own necessity. Mathematics thus appears as an autonomous becoming, governed by necessary sequences, irreducible to any constitutive dependence on the real. It does not coincide with thought, but constitutes its most privileged expression.
Against Ludwig Wittgenstein, Cavaillès maintains the possibility of a theoretical discourse capable of revealing the structure of science. This structure manifests itself in processes of paradigm and thematization, which express the internal dynamics of these sequences. The reference to Baruch Spinoza allows one to conceive this autonomy as a necessary order of ideas. Philosophy thus follows the mathematical movement in order to bring out its necessity, without ever substituting itself for it.
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Notre objectif dans cet article est de prendre la mesure de la formule citée en exergue, et de réfléchir à travers elle sur le problème du rapport de la philosophie aux mathématiques dans l’oeuvre cavaillèsienne. La question sera notamment d’interpréter le « au moyen des mathématiques », figurant dans cette formule. Est-ce à dire que les mathématiques sont proprement pour Cavaillès le cadre d’expression de la pensée ou, comme nous le défendrons ici, qu’elles en constituent une expression privilégiée, plus « parlante » (mais non exclusive) ?
Pour aborder ce problème, nous mettons au jour un double danger : soit le philosophique s’efface dans le mathématique, soit, inversement, le mathématique se trouve soumis à des normes philosophiques extérieures. Dès lors, la difficulté consiste à situer une théorie de la science qui appartienne au mouvement scientifique tout en s’en distinguant. Cavaillès en propose une compréhension comme « auto-illumination » du mouvement scientifique, par laquelle celui-ci laisse apparaître sa nécessité propre. Les mathématiques s’y présentent comme un devenir autonome, régi par des enchaînements nécessaires, irréductible à une dépendance constitutive à l’égard du réel. Elles ne se confondent pas avec la pensée, mais en constituent l’expression la plus privilégiée.
Contre Wittgenstein, Cavaillès maintient la possibilité d’un discours théorique révélant la structure de la science. Cette structure se manifeste dans les processus de paradigme et de thématisation, qui expriment la dynamique interne des enchaînements. La référence à Spinoza permet de penser cette autonomie comme ordre nécessaire des idées. Ainsi, la philosophie suit le mouvement mathématique afin d’en manifester la nécessité, sans jamais s’y substituer.
Our aim in this article is to take the full measure of the formula cited as an epigraph and, through it, to reflect on the problem of the relation between philosophy and mathematics in the work of Jean Cavaillès. In particular, the question will be how to interpret the phrase “by means of mathematics” that appears in this formula. Does this mean that mathematics properly constitutes, for Cavaillès, the framework of expression of thought, or, as we will argue here, that it provides a privileged expression of thought—more “speaking,” though not exclusive?
To approach this problem, we bring to light a double danger: either the philosophical is effaced within the mathematical, or, conversely, the mathematical is subjected to external philosophical norms. The difficulty then consists in situating a theory of science that belongs to the scientific movement while remaining distinct from it. Cavaillès proposes to understand such a theory as an “auto-illumination” of the scientific movement, through which it reveals its own necessity. Mathematics thus appears as an autonomous becoming, governed by necessary sequences, irreducible to any constitutive dependence on the real. It does not coincide with thought, but constitutes its most privileged expression.
Against Ludwig Wittgenstein, Cavaillès maintains the possibility of a theoretical discourse capable of revealing the structure of science. This structure manifests itself in processes of paradigm and thematization, which express the internal dynamics of these sequences. The reference to Baruch Spinoza allows one to conceive this autonomy as a necessary order of ideas. Philosophy thus follows the mathematical movement in order to bring out its necessity, without ever substituting itself for it.
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The youthful writings of Albert Lautman are examined, in particular the Rapport Bouglé of 1935, where the major conceptual nucleus of his subsequent research path aimed at entering 'inside' the contents of the sciences are already identified; it is no coincidence that, from the beginning, the philosophical effort has been directed to understand on the one hand the singularity of mathematics and on the other, to clarify 'les enjeux' of the close connection between mathematics and physics, 'l'unité physico-mathématique'. And all this finds its reasons in Lautman's having been a faithful interpreter of Hermann Weyl and of a certain Hilbert; and developing some of their points has allowed him to give an autonomous contribution to the philosophie mathématique, a chapter of epistemological thought produced in the French-speaking area, unique in its kind and still little known.
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After a brief presentation of the original features of Granger's philosophical system, we propose to show that the notion of mathematical style should not be seen as a substitute for the traditional philosophy of mathematics. To this end, we first show how style depends on the principle of duality. We then try to show, by examining it through the sieve of progress in mathematical logic, that the principle of duality does not keep its promises.
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Après une présentation succinte des traits originaux du système philosophique de Granger, on se propose de montrer que la notion de style mathématique ne peut prétendre être un substitut de la philosophie traditionnelle des mathématiques. À cet effet, on indique d'abord en quel sens le style dépend du principe de dualité. On essaye ensuite de montrer, en le passant au crible des progrès de la logique mathématique, que le principe de dualité ne tient pas ses promesses.
After a brief presentation of the original features of Granger's philosophical system, we propose to show that the notion of mathematical style cannot claim to be a substitute for the traditional philosophy of mathematics. To this end, we first indicate in what sense style depends on the principle of duality. We then attempt to show, by examining it through the lens of advances in mathematical logic, that the principle of duality does not keep its promises.
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Both Bachelard and Vuillemin criticised the Cartesian method, with particular emphasis on the central concept of simple nature. However, despite their apparent convergence, these criticisms turn out to be quite distinct: Bachelard gives these natures an ontological dimension and endows them with absolute self-evidence, while Vuillemin proposes an intuitionist and critical interpretation. We show that the first of these analyses is debatable, and does not really reach Cartesianism, and we measure the philosophical implications of the second, which puts Vuillemin on the track of decisionism.
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Bachelard et Vuillemin ont tous deux critiqué la méthode cartésienne, en insistant plus particulièrement sur le concept central de nature simple. Or, malgré leur apparente convergence, ces critiques s'avèrent bien distinctes : Bachelard donne à ces natures une dimension ontologique et les dote d'une évidence absolue, alors que Vuillemin en propose une interprétation intuitionniste et critique. On montre que la première de ces analyses est discutable, et n'atteint pas réellement le cartésianisme, et on mesure les implications philosophiques de la seconde, qui mettent Vuillemin sur la piste du décisionnisme.
Bachelard and Vuillemin both criticised the Cartesian method, with particular emphasis on the central concept of simple nature. Yet, despite their apparent convergence, these criticisms turn out to be quite distinct: Bachelard gives these natures an ontological dimension and endows them with absolute self-evidence, while Vuillemin proposes an intuitionist and critical interpretation. We show that the first of these analyses is debatable and does not truly reach Cartesianism, and we assess the philosophical implications of the second, which sets Vuillemin on the path of decisionism.
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How can we define a profound mathematical result, and which epistemological criteria should we apply to evaluate the depth of mathematical analysis in the history of mathematics? This article provides a comparative analysis of the works of Gilles-Gaston Granger and Alain Michel on this subject. We argue that the differences in the authors’ perspectives highlight a more fundamental disagreement about how the history of mathematics should be understood. Like Cavaillès, Granger seeks to identify transcendental structures of a kind of creative necessity that are nevertheless incompatible with any form of predictability regarding the evolution of mathematics. In contrast, Michel, along with Brunschvicg and Canguilhem, views mathematical progress as the exploration of a broad range of possibilities. He believes that necessity is merely a retrospective illusion that pays no attention to the details of historical analysis.
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Qu’est-ce qu’un résultat mathématique profond, et quels critères épistémologiques faut-il mettre en œuvre pour caractériser la valeur de la profondeur dans l’analyse de l’histoire des mathématiques ? Cet article propose une analyse comparative des travaux de Gilles-Gaston Granger et d’Alain Michel sur cette question. Nous suggérons que les différences entre les positions de ces deux auteurs sur le thème de la profondeur indiquent un désaccord plus fondamental concernant la conception de l’histoire des mathématiques. Granger, avec Cavaillès, cherche à identifier les structures transcendantales d’une sorte de nécessité créatrice, qui demeurerait toutefois incompatible avec toute forme de prédictibilité de l’évolution des mathématiques. Michel, avec Brunschvicg et Canguilhem, conçoit le progrès mathématique comme l’exploration d’un champ ouvert de possibilités et estime qu’il n’est possible d’y voir un développement nécessaire qu’avec un regard rétrospectif, indifférent aux détails de l’analyse historique.
How can we define a profound mathematical result, and which epistemological criteria should we apply to evaluate the depth of mathematical analysis in the history of mathematics? This article provides a comparative analysis of the works of Gilles-Gaston Granger and Alain Michel on this subject. We argue that the differences in the authors’ perspectives highlight a more fundamental disagreement about how the history of mathematics should be understood. Like Cavaillès, Granger seeks to identify transcendental structures of a kind of creative necessity that are nevertheless incompatible with any form of predictability regarding the evolution of mathematics. In contrast, Michel, along with Brunschvicg and Canguilhem, views mathematical progress as the exploration of a broad range of possibilities. He believes that necessity is merely a retrospective illusion that pays no attention to the details of historical analysis.
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The philosophy of the history of mathematics expounded by Cavaillès bears a close and contrasting relationship to that set out by Brunschvicg in La Modalité du Jugement. (1897). The activity of scientific judgement is said to be mixed, between (ideal) judgements of interiority and (realistic) judgements of exteriority. The mixed form of the historical activity of knowledge is the modality of the possible. Hence a historical epistemology that claims Kantian idealist filiation and rejects speculative idealism. Cavaillès, a thinker of the creative necessity of mathematical development, reduces the role of intellectual adventure and possibility in its history, and in so doing distances himself from a master to whom Canguilhem would have remained closer. The recent history of mathematics, while paving the way for Kronecker's revenge on the abstract set-theoretical conceptions that inspired Cavaillès's necessitarianism, leads us to reconsider the radicalism that opposed him to Brunschvicg's philosophy of the modality of judgement.
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Using examples from number theory, this paper explores the philosophical notion of mathematical “depth” — what makes certain results more profound or significant than others. English translation of « Mathématiques et ‘profondeur’ : l’exemple de la théorie des nombres » (Jean-Toussaint Desanti, une pensée et son site, ed. G. Ravis-Giordani, Fontenay-aux-Roses: ENS Éditions, 2000, pp. 181–199).
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A tribute to Pierre Cartier, one of the last universal mathematicians, member of Bourbaki, and a lifelong advocate of the dialogue between mathematics and philosophy.
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A tribute to Pierre Cartier, one of the last universal mathematicians, member of Bourbaki, and a lifelong advocate of the dialogue between mathematics and philosophy.
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Some personal and intellectual notes on the life and work of F. William Lawvere, the founder of categorical logic and a deep philosophical thinker about mathematics.
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