Some remarks on Husserl’s Doppelvortrag

We provide an interpretation of Husserl’s 1901 Doppelvortrag in Göttingen from the viewpoint of the modernist transformation of mathematics. We emphasise the dialectical aspects of the Doppelvortrag , and especially the underlying conflict between abstraction and intuition , which often resurfaces in Husserl’s philosophy. We focus on three key aspects: (1) the relation between Husserl’s idea of pure logic and Hilbert’s axiomatic method; (2) the concept of Definitheit and the early developments of model theory; (3) the nature of imaginary numbers and their justification in arithmetic and algebra. We stress that, in contrast to Hilbert’s Completeness Axiom from his Grundlagen , Husserl viewed the Definitheit as a statement requiring a proof, and he believed that one could be provided in the cases of arithmetic and geometry.