 The importance of a rigorous account of any part of mathematics should not be ignored, but the acceptance of non-Euclidean and Riemannian geometries went beyond the presentation of a consistent formalism. It marks the acceptance of the abstract view that geometry is whatever can be described in the Riemannian formalism: one has a very general framework, allowing for a dizzying number of concrete specifications. Thus the door was opened to the view that there are many geometries, each of which must be consistent, and none of which need to refer to Euclidean space, however intuitive this may be. (A parallel development with similar outcome happened in the context of projective geometry with Klein's work.) The number of dimensions of the ‘space’ under discussion, and the precise metric can be modified at will (in due course, the topological character of that ‘space’ would be treated similarly). There is a 2-dimensional geometry of such-and-such a kind because a suitable metric can be found; because there is, as it were, a map of it, not because a surface has been found in \(\mathbb{R}^3\) with the right properties. Indeed, it was later shown (Hilbert 1901) that there is no surface in \(\mathbb{R}^3\) corresponding precisely to non-Euclidean 2-dimensional space.