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Journal Article

Carnap’s Metrical Conventionalism versus Differential Topology

Thomas Mormann
Philosophy of Science
Vol. 72, No. 5, Proceedings of the 2004 Biennial Meeting of The Philosophy of Science AssociationPart I: Contributed PapersEdited by Miriam Solomon (December 2005), pp. 814-825
DOI: 10.1086/508112
https://www.jstor.org/stable/10.1086/508112
Page Count: 12
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Carnap’s Metrical Conventionalism versus Differential Topology
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Abstract

Geometry was a main source of inspiration for Carnap’s conventionalism. Taking Poincaré as his witness, Carnap asserted in his dissertation Der Raum ([1922] 1978) that the metrical structure of space is conventional while the underlying topological structure describes ‘objective’ facts. With only minor modifications he stuck to this account throughout his life. The aim of this paper is to disprove Carnap’s contention by invoking some classical theorems of differential topology. It is shown that his metrical conventionalism is indefensible for mathematical reasons. This implies that the relation between topology and geometry cannot be conceptualized as analogous to the relation between the meaning of a proposition and its expression in some language as the logical empiricists used to say.