The Epistemology of Geometry

Abstract

There is a philosophical tradition, going back at least to Poincare ([9]), which argues that the geometrical features of the universe are underdetermined by all possible evidence, by all of the actual or possible coincidences and trajectories of material things, whatever they may be. Many different geometrical and physical theories can encompass the phenomena, can account for the motions of things. Poincare supported his view with a parable, Reichenbach ([11]) with a sort of recipe for writing down alternative but empirically equivalent theories; later authors have repeated their arguments or given very similar ones. In his admirable book on space, time, and spacetime ([13] ), Lawrence Sklar has tried to catalogue the possible philosophical attitudes towards the underdetermination arguments put forward by Poincare, Reichenbach and others: One can simply be sceptical about the possibility of knowing geometrical truths; one can maintain that in so far as there are any such, they are truths by convention; one can contend that certain theories are a priori more plausible than others and so should win any ties based on empirical evidence; one can insist that despite appearances all empirically equivalent theories say the same thing; or one can deny that there is any coherent notion of empirical equivalence and so lay the entire question aside. What one cannot do, if this catalogue of options is complete, is to admit the notion of empirical equivalence, admit an account of sameness of meaning which permits that different theories may save the phenomena, deny that there are available a priori principles about what is most likely true, and still insist

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