Synthese (2007) 156:281–310 DOI 10.1007/s11229-006-0008-y RESEARCH ARTICLE Towards completeness: Husserl on theories of manifolds 1890–1901 Mirja Helena Hartimo Received: 23 March 2005 / Accepted: 22 March 2006 / Published online: 31 January 2007 ©Springer Science+Business Medi B.V. 2007 Abstract Husserl’s notion of definiteness, i.e., completeness is crucial to understand- ing Husserl’s view of logic, and consequently several related philosophical views, such as his argument against psychologism, his notion of ideality, and his view of formal ontology. Initially Husserl developed the notion of definiteness to clarify Hermann Hankel’s ‘principle of permanence’. One of the first attempts at formulating defi- niteness can be found in the Philosophy of Arithmetic, where definiteness serves the purpose of the modern notion of ‘soundness’ and leads Husserl to a ‘computa- tional’ view of logic. Inspired by Gauss and Grassmann Husserl then undertakes a further investigation of theories of manifolds. When Husserl subsequently re- nounces psychologism and changes his view of logic, his idea of definiteness also develops. The notion of definiteness is discussed most extensively in the pair of lectures Husserl gave in front of the mathematical society in Göttingen (1901). A detailed analysis of the lectures, together with an elaboration of Husserl’s lectures on logic beginning in 1895, shows that Husserl meant by definiteness what is today called ‘categoricity’. In so doing Husserl was not doing anything particularly orig- inal; since Dedekind’s ‘Was sind und sollen die Zahlen’ (1888) the notion was ‘in the air’. It also characterizes Hilbert’s (1900) notion of completeness. In the end, Husserl’s view of definiteness is discussed in light of Gödel’s (1931) incompleteness results. Keywords Early Husserl · Definiteness · Completeness · Theories of manifolds M. H. Hartimo (B ), Department of Mathematics, Statistics and Philosophy, University of Tampere, 33014, Finland e-mail: Mirja.Hartimo@uta.fi