Abstract
It has been said that it was Reichenbach’s merit to have realized that the conventional aspects of geometry are precisely the metrical aspects of geometry. I disagree. The most suggestive parts of Reichenbach’s work on space and time (1958) are his remarks on the possibility of alternative topologies for physical space. For, if Reichenbach is correct, the topology of space has an intimate connection with both the causal structure of the universe and with the identity of objects in time. We have alternatives for the first just because we have alternatives for the latter two. My intent is to apply Reichenbach’s insights on the relativity of topology, and some of Felix Klein’s work on the connections between local geometry and global geometry, to the subject of relativistic cosmology.
Aus der Annahme, dass der uns umgebende Raum eine euklidische oder hyperbolische Struktur aufweist, lässt sich keinswegs folgern, dass dieser Raum eine unendliche Aus-denung besitzt; denn die euklidsche Geometrie ist z.B. durchaus mit der Annahme einer endlichen Raumausdehnung verträglich, eine Tatsache, die man früher übersehen hat. Diese Möglichkeit, dem Weltall auch bei beliebiger Struktur einen endlichen Inhalt zuzuschrieben, ist besonders wertvoll, weil die Vorstellung einer unendlichen Ausdehnung, die zunächst als wesentlicher Fortschritt des menschlichen Geistes betrachet wurde, mannigfache Schwierigkeiten, z.B. bei dem Problem der Massenverteilung, mit sich bringt. Man sieht hieraus, wie tief alle diese Überlegungen in die kosmologischen Probleme eingreifen. F. Klein (1928)
This paper has benefited from conversations and correspondence with John Earman, Richard Grandy, Gilbert Hartman, Hartry Field, and Michael. I am indebted to Bas van Fraassen for reading preliminary versions and encouraging their fefinement.
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Glymour, C. (1973). Topology, Cosmology and Convention. In: Suppes, P. (eds) Space, Time and Geometry. Synthese Library. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2686-4_10
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