Abstract

Are the Euclidean axioms false of physical configurations? Do we answer this by empirical observation? On the prevailing view both answers are affirmative, and the empirical subject matter of geometry is "physical space." I dispute this view on conceptual grounds, beginning with an analysis of place-relational concepts and arguing that these are fundamental to our understanding of the empirical world as a spatio-temporally ordered system of objects and events, and to the observer's perceptual and cognitive orientation, which physical theory, grounded in empirical observation, presupposes, and to which statements of the methods and objectives of empirical science appeal. I then argue that attempts by Nagel and Reichenbach to show how observation can count against Euclid equivocate, and confuse the problem of justification with the problem of meaning; that an adequate understanding of geometry must acknowledge the observational meaning conditions of its nonlogical terms ; that Euclidean axioms are true of physical configurations only if their nonlogical terms are interpreted accordingly; that the truth value of geometric axioms is determined by analysis of geometry's proper subject matter: place-relational concepts; that the terms "non-Euclidean space" and "curved space," unless used to refer to surfaces, may embody confusion, because "curved" is properly applied to spatial objects, coordinate systems, and surfaces, not to three-dimensional space; that proper analysis of the infinitesimal method of tensor calculus, which rests on Euclidean methods, shows that the use of Riemannian analytic geometry does not support the non-Euclidean thesis as Sklar and others claim, particularly since Euclidean axioms purport to be applicable to two- and three-dimensional sets of place relations, not Minkowski spacetime of Special Relativity, or Riemannian spacetime of General Relativit y. Finally, I argue that, insofar as the relativistic concept of a spacetime interval requires explanation by appeal to the pre-relativistic, observationally grounded concepts of spatial distance and temporal interval, these latter concepts are more basic than the former