From Classical to Modern Mathematics: Hilbert’s Formalism and Kant’s Schematism

Abstract

 This essay examines the manner in which Hilbert worked out his first formalism in his investigations on the foundations of geometry. To elucidate these views, Hilbert’s position is compared with that of Kant, who set forth a constructive notion of “geometrical objects” which endured in the Philosophy of mathematics for a long time. In the first part, the author explores the way in which Kant explains the notion of proof in classical geometry and clarifies how his account relates to his theory of schematism. Next, the conception underlying Hilbert’s Grundlagen der Geometrie is presented and an attempt is made to recreate the path he followed until he reached his point of view. In particular this article explores the role that projective geometry and the principle of duality played in his reflections. Finally, Kant’s ideas are contrasted with those of Hilbert in his first formalism, pointing toward the view that the latter constitutes a necessary generalization of Kant’s mathematical philosophy.