Geometry

There, curiously, Riemann's ideas remained for more than a generation. There was some interest in novel three-dimensional geometries, almost none in geometries, in Riemann's sense, of higher dimensions, except to show that mechanics could be done in such a setting, and in simplifying the formidable algebraic complexity of the subject (today handled by means of the tensor calculus). The decisive change came with the work of Albert Einstein (1879–1955).

Einstein's special theory of relativity of 1905 was a thorough reworking of the mathematics of motion, and at first Einstein was unsympathetic to the geometrical reworking given to his ideas by Hermann Minkowski (1864–1909) in 1908. But when Einstein started to think of a general theory of relativity that would find an equivalence between forces and accelerations, he found the ideas of Riemannian differential geometry invaluable. The theory he came to in 1915 formulated gravitation as a change in the metric of space-time. On this theory, matter changes the shape of space, which is how it exerts its attractive force.

Felix Klein's Erlangen program.

By the 1870s, projective geometry had established itself as the fundamental geometry, with Euclidean geometry as a special case, along with some other geometries not described in this essay. The young German mathematician Felix Klein (1849–1925) then proposed to unify the subject, by treating projective geometry as the main geometry, and deriving the other geometries as special cases. Every geometry Klein was interested in, most strikingly non-Euclidean geometry, was defined on projective space or a subset of it, and the relevant geometric properties were those kept invariant by the action of a suitable subgroup of the group of all projective transformations. This view, called the Erlangen Program, after the university where Klein first published it, has remained the orthodoxy since the 1890s, when Klein republished it, but in its day the novelty was the explicit introduction of the then-novel concept of a group, and the shift of attention from properties of figures to the idea that these properties are geometric precisely because they are invariant under the appropriate group of transformations.

Italians, Hilbert, and the axiomatization of geometry.

Klein's geometries do not include many of the geometries Riemann had pointed toward. It included only those that have large groups of transformations, which, however, is most of those of interest in physics and much of mathematics for a long time, including, it was to transpire, Einstein's special theory of relativity. The first step beyond what Klein had done, and for that matter Riemann, was proposed by David Hilbert (1862–1943), starting in 1899, although a number of Italian geometers had had similar ideas in the decade before.

Hilbert was insistent that theorems in geometry should only use the properties of objects that entered into their definitions, and to this end he formulated elementary geometry carefully in terms of five families of axioms. What distinguished his work from his Italian predecessors was his insistence that there was an interesting new branch of mathematics to be explored, which studied axiom systems. It would aim at showing the independence of certain axioms from others, and establishing what sorts of axioms were needed to deduce particular results. Whereas the Italian mathematicians had mostly aimed at axiomatizing projective and perhaps Euclidean geometry once and for all, Hilbert saw the axiomatic method as both creative and of wide applicability. He even indicated ways in which it could work in mathematical physics.

By 1915, the axiomatization of geometry had begun to spread to other branches of mathematics as well, with a consequent improvement in the standards of formal proof, and Einstein's general theory of relativity had similarly improved the ideas about the applications of geometry.