On the occasion of the 150th birthday of Georg Cantor (1845–1918), the founder of the theory of sets, the development of the logical foundations of this theory is described as a sequence of catastrophes and of trials to save it. Presently, most mathematicians agree that the set theory exactly defines the subject of mathematics, i.e., any subject is a mathematical one if it may be defined in the language (i.e., in the notions) of set theory. Hence the nature of formal (...) definitions plays an important role within the logical foundations of mathematics. Its study is also helpful to answer the question of how it is possible that the set theory as a universal new ontology for the subject of mathematics (as people hoped around 1900) totally failed but nevertheless the language of set theory is successful in all the mathematical practice. (shrink)

According to the received view, Russell rediscovered about 1900 the logical definition of cardinal number given by Frege in 1884. In the same way, we are told, he stated and developed independently the idea of logicism, using the principle of abstraction as the philosophical ground. Furthermore, the role commonly ascribed in this to Peano was only to invent an appropriate notation to be used as mere instrument. In this paper I hold that the study of Russell's unpublished manuscripts and Peano's (...) (and disciples) writings (as part of a larger investigation only pointed out here) shows, on the contrary, that Russell obtained the method to transform definitions by abstraction into nominal definitions and the general logicist idea from Peano (and his school). The only original insight from Russell, the principle of abstraction, partially derived from Moore's early philosophy, was finally abandoned because it was not of practical use. (shrink)

Few have entertained the idea that Georg Cantor, the creator of set theory, might have influenced Edmund Husserl, the founder of the phenomenological movement. Yet an exchange of ideas took place between them when Cantor was at the height of his creative powers and Husserl in the throes of an intellectual struggle during which his ideas were particularly malleable and changed considerably and definitively. Here their writings are examined to show how Husserl's and Cantor's ideas overlapped and crisscrossed in the (...) areas of philosophy and mathematics, arithmetization, abstraction, consciousness and pure logic, psychologism, metaphysical idealism, new numbers, and sets and manifolds. (shrink)

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)