In November 2010, following an invitation by Claudio Majolino, I gave a presentation in the seminar Formes et Trasformations in Lille, on “Husserl’s Notion of Manifold: Beyond Cantor and Riemann”.
In my presentation I argued that the concept of Mannigfaltigkeit (variously translated as “manifold” or “multiplicity”) 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 exclusively in the context of Husserl’s early period in Halle, while he was elaborating his habilitation essay into the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor. However, at the time Husserl distanced himself explicitly from Cantor’s definition and was rather inspired by Bernhard Riemann’s work, having studied and lectured extensively on Riemann’s theories of space. Cantor generally used “manifold” simply as synonym for “Menge” (quantity), or “Inbegriff” (collection), thereby laying the foundations for set-theory. Indeed, in the mid 1890s he started calling his work Mengenlehre (“theory of quantities”), instead of Mannigfaltigkeitslehre (“theory of manifolds”).
Husserl is well aware of the differences between Cantor and Riemann, e.g. in the period of the Philosophy of Arithmetic he tells us:
By manifold, Cantor means a simple collection of elements that are in some way united. [. . . ] However, this conception does not coincide with that of Riemann and as used elsewhere in the theory of geometry, according to which a manifold is a collection not of merely united, but also ordered elements, and on the other hand not merely united, but continuously connected elements. (Hua XXI, p. 95 f.)
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 we actually already have in mathematics, this sounds slightly odd and is a bit misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis.
Mathematics, according to its highest idea, is the theory of theory [Theorienlehre], the most general science of the possible deductive systems in general (Hua CW X, p. 411)
Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will later 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 discuss 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.
The presentation has now been published in a French translation as “La notion husserlienne de multiplicité : au-delà de Cantor et Riemann” in Methodos and is freely available on-line.