Schopenhauer and the Mathematical Intuition as the Foundation of Geometry

Abstract

Schopenhauer did not write extensively on mathematics, but he discussed the subject in almost all of his works. His thesis about the superiority of intuition in establishing the truth of geometrical theorems became a battle against the traditional demonstrative procedure in geometry. Commentators have generally provided internal readings of Schopenhauer’s texts on mathematics but have neglected their context. This paper examines Schopenhauer’s philosophy of mathematics by discussing its relationship with both his views on the acquisition of knowledge and his familiarity with the contemporary British discussions of mathematics. An overview of his ideas on the primacy of intuition in both mathematics and its teaching is the basis of this inquiry into the connection of those ideas with both his conception of the role of mathematics in natural philosophy and his encounter with the 1830s British texts on mathematics, which he quoted in the second volume of The World as Will and Representation. By making him aware that Euclidean geometry required a thorough scrutiny of its foundations—notwithstanding its undisputed reputation—these texts contributed to the hitherto unappreciated modifications in his mathematical considerations. Schopenhauer participated in an early phase of the debate on the foundations of geometry by taking a fresh look at intuition: not only as an alternative to demonstration, but also as the ground of truth and certainty in the Euclidean system.