Poincare's Thesis of the Translatability of Euclidean and Non-Euclidean Geometries

Abstract

Poincare's claim that Euclidean and non-Euclidean geometries are translatable has generally been thought to be based on his introduction of a model to prove the consistency of Lobachevskian geometry and to be equivalent to a claim that Euclidean and non-Euclidean gemoetries are logically isomorphic axiomatic systems. In contrast to the standard view, I argue that Poincare's translation thesis has a mathematical, rather than a meta-mathematical basis. The mathematical basis of Poincare's translation thesis is that the underlying manifolds of Euclidean and Lobachevskian geometries are homeomorphic. Assuming as Poincare's does that metric relations are not factual, it follows that we can rewrite a physical theory using Euclidean geometry as one using Lobachevskian geometry and express the same facts. Poincare also claims that the converse is true and, because of his desire to prove that science is cumulative, it is the converse that is most important to him (1982, p. 140). He holds that we could continue to use Euclidean geometry in physics no matter what, sohe must be able to rewrite in a Euclidean framework any experiment which seems to prove that the world is Lobachevskian. Poincare's conventionalism depends on the possibility of a relational dynamics which goes further than currently accepted theories, so we must admit that his defense of his conventionalist thesis is inadequate. However, we are provided with a plausible mathematical