Paul Cortois

In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of its complementary dimensions: paradigmatic generalization (domain extension, descriptive definitions, creative role of the symbolism...) and thematic reflexivity of concepts (promotion of operations to objects of a higher type).

RésuméCette étude essaie de réunir quelques éléments en vue d'une interprétation plus complète du programme épistémologique contenu de façon extrêmement elliptique dans l'ouvrage post‐hume de Jean Cavaillès, publié en 1947 par G. Canguilhem et C. Ehresmann sous le titre Sur la logique et la théorie de la science. Afin d'arriver, plus particulière‐texte, où la «nécessité génératrice» d'une dialectique conceptuelle est évoquée, il faudra rappeler quelques aspects connus ainsi que quelques aspects moins connus du cheminement philosophique de Cavaillès.Cette reconstruction est mise en œuvre de façon détaillée dans un travail de commentaire textuel qui devait permettre de dégager le sens des formules denses et souvent allusives de Cavaillés dans son dernier ouvrage1. Il n'est pas possible de résumer ici en quelques lignes le contenu d'un tel travail. Toutefois on peut indiquer quelques conclusions qui s'en dégagent pour l'interprétation globale de l'ouvrage en question. En premier lieu il faut reconnaître la nature même du problème traité ici par Cavaillès. En second lieu on caractérise le type de position prise envers ce problème ainsi défini et en tant qu'on peut le restreindre dans un premier moment à la philosophie des mathématiques. En troisième lieu on montre que se problème subit un changement d'éclairage par le fait que cette restriction doit être levée en prenant en considération le rapport entre expérience mathématique et expiérience physique. que nous serons obligés d'entrer un peu plus, et à titre d'exemple, dam le détail de l'exégèse). Finalement on peut essayer de préciser le programme d'une dialectique conceptuelle en quelques thèses ou titres schématiques qui concement les questions majeures de la théorie de la science selon Cavaillès2.

The arrow of time has been invoked to bridge all gaps between the 'two cultures'. Would time also help to mediate between the sphere of cognition (epistemic meaning) and the sphere of Bedeutsamkeit (meaning-as-relevance) when taking ritual to be a strongly idiosyncratic representative of the latter? What is the role of time in the modes of meaning in the realm of scientific concepts in their most rigorous shape (the mathematical) on the one hand, in ritual on the other hand? Taking an external point of view, surprising connections can be laid, especially when focussing on the relations of form and materiality or content in the respective fields. Ritual can be considered — according toits formal structure — as an embodiment of the notion of an algorithm. When looking at the temporal structure of the respective symbolic practices from an internal point of view however, their contrasting modes of creating meaning become irreducible. In particular, the algorithmic-sequential mode in ritual is a shell for another time, constitutive of a proper Bedeutsamkeit. This is analyzed in terms of the role of the past in ritualized meaning (and illustrated via the art of time in the Japanese tea ceremony). Whereas a paradoxical creation of a past as something lost and nevertheless to be recollected is constitutive for the meaning of the ritual gesture, the proper element of the concept is time's arrow always pointing forward, to the next stage. Here the past, while embodied in the scientist's gesture as trace of epistemic styles and traditions, at the same time has to be suppressed in the interest of the creation of the new.

Whatever the precise analysis of the notion of an `internal point of view', to talk about `religious traditions' is to imply that traditions of a certain kind primarily deploy an internal point of view. But what can be said about the notion of an intellectual tradition that would at the same time also be, or be connected to, a religious tradition? To some, such notions appear to border on contradiction. In accordance with the Cartesian criticism of coutume et exemple, we tend to think of an intellectual stance as the correlate of a detached view, with the rational remaining the enemy of the traditional. In that case, every application whatsoever of the predicate `intellectual' to a tradition would be flawed. According to a certain Enlightenment view, inspired by science, is it not of the very essence of the intellectual stance in general that one stand back at a critical distance from any tradition, especially one's own?In fact, historically oriented philosophers of science like Crombie, Kuhn, Laudan have reminded us of the fact that there is nothing disturbingly paradoxical in speaking of `intellectual traditions': to quote one of Kuhn's book titles, there is the "essential tension" that characterizes science in its development just as much as any other cultural tradition. This tension between the old and the new is the mark of traditions, including the ones defining the rationality or rationalities of science. We could say that Kuhn's `paradigms' already outlined the borders of specific `catholicities' for the specialized fields of scientific research traditions. If Kuhn ever had a point, it must be that science too can develop only within the framework of styles of reasoning, which is to say that it demands viewpoints that cannot be characterized as `external', at least not in an absolute, metaphysical sense of the word.