Grünbaum on the Conventionality of Geometry

Abstract

Physical geometry is an especially good case study for one interested in the problem of conventionalism in science for at least two reasons. First, there exist a large number of alternative geometrical theories which have all been extensively investigated. This contrasts sharply with most other areas of science in which ‘alternatives’ are largely conjectural. Thus, in the case of geometry, claims that our theories are in some sense conventional or arbitrary, in that we could just as well have had others, at least have more substance. Second, the issue of the conventionality of geometry appears to depend on one particular problem: the status of our methods of measuring length or our ‘congruence definitions’. Whether physical geometry is in some sense arbitrary or conventional depends on whether there is a sense in which our methods of measurement or our ‘congruence definitions’ are arbitrary or conventional. This, of course, simply reflects the mathematical fact that all the properties of a geometry are determined once a metric is determined.

I am indebted to Clark Glymour for correcting several mistakes in my argument.