Pour projeter de la lumière dans de nombreux coins et recoins obscurs de la logique pure de Husserl et dans les rapports entre sa logique formelle et sa logique transcendantale, et combler des lacunes empêchant qu’on arrive à une appréciation juste de sa Mannigfaltigkeitslehre, ou théorie de multiplicités, on examine comment, en prônant une théorie des systèmes déductifs, ou systèmes d’axiomes, comme tâche suprême de la logique pure, Husserl cherchait à résoudre certains problèmes épineux auxquels il s’était heurté en (...) écrivant Philosophie de l’arithmétique. Ces problèmes sont décrits. Ensuite, on rassemble les éléments nécessaires pour caractériser ce que Husserl, à travers les textes présentement disponibles, voulait dire des Mannigfaltigkeiten. Pour conclure, il est indiqué comment Husserl pouvait considérer que sa théorie représentait une solution aux problèmes qui avaient conduit à son élaboration.To shed light on the numerous dark areas surrounding Husserl’s ideas about pure logic and the relationship between his formal logic and his transcendental logic, and so provide an accurate characterisation of his Mannigfaltigkeitslehre, a close look is taken at how, by advocating a theory of deductive systems, or axiomatic systems as the highest task of pure logic, Husserl sought to resolve certain thorny problems he encountered when writing the Philosophy of Arithmetic. Those problems are described. Then, his ideas about axiomatization and manifolds are drawn together from the various sources now available to provide a more complete picture of his Mannigfaltigkeitslehre. In conclusion, it is shown how Husserl considered his theory to be a solution to the problems leading to its development. (shrink)

Este artigo problematiza a concepção de mathesis universalis na filosofia de Edmund Husserl a partir de uma discussão sobre a noção de Mannigfaltigkeitslehre. Inicialmente, retoma-se a questão dos números “imaginários” e depois é repassado o contexto matemático que influenciou o desenvolvimento filosófico husserliano. Finalmente, a concepção de mathesis universalis é enquadrada nas referências históricas de Descartes e Leibniz para, finalmente, mostrar as implicações fenomenológicas próprias do pensamento de Husserl.

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of space. Husserl’s (...) Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development. (shrink)

C'est un thème husserlien que je veux traiter dans mon exposé, un thème qui relève de la philosophie husserlienne de la logique. Celui de la «Mannigfaltigkeitslehre» ou doctrine de la multiplicité. J'exposerai brièvement ce thème et j'essaierai ensuite d'en donner une version moderne qui soit à la fois une critique et un renouvellement des intentions de Husserl.

C'est un thème husserlien que je veux traiter dans mon exposé, un thème qui relève de la philosophie husserlienne de la logique. Celui de la «Mannigfaltigkeitslehre» ou doctrine de la multiplicité. J'exposerai brièvement ce thème et j'essaierai ensuite d'en donner une version moderne qui soit à la fois une critique et un renouvellement des intentions de Husserl.