From Grassmann, Riemann to Husserl: a brief history of concept of Manifold

Abstract

Edmund Husserl’s theory of manifold (Mannigfaltigkeitslehre) was formalized for the first time in his Philosophie der Arithmetik; in his Logische Untersuchungen, §§69–70; also discussed in Ideen I, §§72; in Formale und Trascendentale Logik, §§51–54; in Logik und allgemeine Wissenschaftstheorie, chapter two; and finally it appears in Einleitung in die Logik und Erkenntnistheorie, §§18–19. In each of these books, Husserl presents a concept of manifolds as an ontological form. Such form is necessarily axiomatic and appears as inspired by Bernhard Riemann’s work. Indeed, Husserl, who studied and lectured extensively on Riemann’s theories of space, presented his own conception of mathematics as a theory of manifolds as a generalization of Riemann’s notion of manifold.

Keywords: Husserl, Riemann, Grassmann, Manifold, Phenomenology