Structuralist themes in the early Husserl

Clinton Tolley

Uuniversity of California at San Diego

Erich Reck

University of California

Abstract: 
In the late 1890s and early 1900s, Husserl devoted considerable efforts toward developing a “theory of manifolds [Mannigfaltigkeitslehre]”. There are evidently ties to developments in 19th-century mathematics, e.g., Bernhard Riemann’s novel approach to geometry. In the 1900 Prolegomena to his Logische Untersuchungen, Husserl describes what he is after as “a science that deals apriori with the essential sorts (forms) of theories and the relevant laws of relation”, a science in which “the essential concepts and laws which belong constitutively to the idea of theory will be investigated” (§69), i.e., as a general theory of theories. Husserl’s discussions in this connection have been the subject of a fast growing secondary literature. That literature has focused especially (and understandably) on establishing points of kinship and contrast between his concern for what he calls the definiteness of the manifolds investigated by logic and mathematics, on the one hand, and David Hilbert’s contemporaneous discussions of completeness in meta-mathematics, on the other hand (cf. Majer 1997, Da Silva 2000, Hartimo 2007, Hill 2010, Centrone 2010). What has been of less direct concern so far is how Husserl’s remarks on manifolds connect up with his broader concerns for a universal system of logic and ontology. Yet in the Prolegomena (§67), Husserl claims that the pure theory of manifolds must be preceded by an investigation of the “pure categories of meaning [Bedeutungskategorien]” (e.g., the categories: concept, proposition, truth, etc.), or what he calls a “pure grammar”, as well as by an investigation of the “pure objectual categories [gegenst¨andlichen Kategorien]” (e.g., the categories: object, state of affairs, relation, connection, etc.), so “pure ontology”. Our presentation aims, then, to extend existing interpretations by focusing especially on the ontological underpinnings of Husserl’s discussion of the theory of manifolds. We will explore, in particular, the extent to which Husserl’s remarks would seem to anticipate directly current structuralist accounts of the ontology of mathematics (as has been suggested in Parsons 2008). For Husserl claims in the Prolegomena (§70) that, from the point of view of the theory of manifolds, each manifold is a “domain which is uniquely and solely determined” by “falling under” a theory of the relevant form, with the “objects” in question “remaining quite indefinite as regards their matter”. Such objects “are not determined directly as individual or specific singulars, nor indirectly by way of their material species or genera, but solely by the form of the connections attributed to them”. Our goal will be to unpack these and other suggestive remarks Husserl makes at the time (e.g., in his 1901 “Double Lecture” on axiomatic systems) about the ontology underlying his theory of manifolds, by placing them with the context of his broader ontological classification-scheme, most notably in the Logische Untersuchungen. Here we will focus especially on Husserl’s doctrines of universal objects and abstract individuals, and on his general mereology, respectively. We will thus be able to ascertain how well Husserl’s categories can be put into coordination with his more well-known structuralist contemporary, Richard Dedekind, as well as with those employed in more recent structuralisms.