Abstract

Geometric statements are expressions of natural language with descriptive meaning, since they refer to things such as triangles, and to their characteristics. It is shown, following statements by authors such as Carnap (Philosophical foundations of physics) that geometric meanings are not factual, but mathematical, so that it is not geometric theories that apply, but geometrically interpreted factual theories. Under the Euclidean paradigm, the object of study of geometry was space, but the theoretical pluralism that replaced it resulted in the creation in the discipline of a new plural ontology according to which there are different spaces: Euclidean, hyperbolic and elliptic. Thus, geometrical knowledge no longer seems to be a knowledge of the universal (of universal characteristics of space), but of the particular: of individual spaces. If geometrical theories are mathematical, these questions constitute obstacles both to the thesis that mathematical knowledge is universal, and to the thesis that it is universally applicable.