A Constructive Definition of Space via Dynamical Evolution and Observational Acts

Abstract

We propose a constructive and dynamical redefinition of spatial structure, grounded in the interplay between mechanical evolution and observational acts. Rather than presupposing space as a static background, we interpret space as an emergent entity that arises through observational acts. Given a time evolution and a physically specified measurement context, we construct a reachability structure and a context-dependent measurable structure. Using the framework of pre-topologies, measurable structures, and the GNS construction, we analyze how the choice of observables and the system's time evolution dynamically determine the topological and measure-theoretic features of space. Time evolution and the measurement context are inputs to this construction. Deriving time itself or a complete dynamics for the selection and stabilization of the measurement context lies outside the scope of this paper. This approach highlights the observer-dependent and context-sensitive nature of spatial concepts in both classical and quantum domains.

32 pages, Clarified notation in Sec. 2.1, added discussion on point-less geometry in noncommutative geometry, and included remarks on phase-space-first perspectives in quantum gravity. v3. Revised after peer review. Accepted for publication in Annals of the Japan Association for Philosophy of Science, Vol. 37 (2028)