Die Verortung der Mathematik in Hegels System: Die Bestimmung des Verhältnisses von Arithmetik und Geometrie anhand des Verhältnisses von logischer Zahl und naturphilosophischem Raum
Achim Wamßler
Hegel-Studien Band 57 (2024)

Abstract: This essay concerns the question, why Hegel discusses the mathematical subdisciplines of arithmetic and geometry at different places within his system. As an introduction, I argue that this is unusual compared to Hegel’s treatment of other sciences. Furthermore, I discuss four possible models of how mathematics as a whole is located in Hegel’s philosophy. I then argue for the most plausible of these models by first elaborating a structural parallel between Hegel’s discussion of the concept of number in his Logic and his development of the concept of space at the beginning of his Philosophy of Nature: For Hegel, both number and space have the determinations of continuity, discreteness, and externality. After this, I discuss the specific difference between Hegel’s concept of number and his treatment of space. It consists of the fact that space is necessarily related to intuition, while number is not. In a final step, I discuss various possibilities of the relationship between number and space in order to demonstrate that both concepts (and thus Hegel’s foundations of arithmetic and geometry) – despite the different locations in which they are addressed – are understood by Hegel as part of a coherent mathematical science. I conclude that the most plausible relationship between number and space is one in which the concept of number conceptually grounds Hegel’s geometrical understanding of space.

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