Space-time Geometry Translated into the Hegelian and Intuitionist Systems

Abstract

Kant noted the importance of spatial and temporal intuitions (synthetics) in geometric reasoning, but intuitions lend themselves to different interpretations and a more solid grounding may be sought in formality. In mathematics David Hilbert defended formality, while L. E. J. Brouwer cited intuitions that remain unencompassed by formality. In this paper, the conflict between formality and intuition is again investigated, and it is found to impact on our interpretations of space-time as translated into the language of geometry. It is argued that that language as a formal system works because of an auxiliary innateness that carries sentience, or feeling. Therefore, the formality is necessarily incomplete as sentience is beyond its reach. Specifically, it is argued that sentience is covertly connected to space-time geometry when axioms of congruency are stipulated, essentially hiding in the formality what is sense-certain. Accordingly, geometry is constructed from primitive intuitions represented by one-pointedness and route-invariance. Geometry is recognized as a two-sided language that permitted a Hegelian passage from Euclidean geometry to Riemannian geometry. The concepts of general relativity, quantum mechanics and entropy-irreversibility are found to be the consequences of linguistic type reasoning, and perceived conflicts (e.g., the puzzle of quantum gravity) are conflicts only within formal linguistic systems. Therefore, the conflicts do not survive beyond the synthetics because what is felt relates to inexplicable feeling, and because the question of synthesis returns only to Hegel’s absolute Notion.