Newton’s absolute space as an arena for physical processes constituted an inherent element of the mechanistic worldview, and it came as a shock when it turned out that Euclidean space is not the only possibility. The dispute concerning Euclid’s “fifth postulate” lasted from antiquity. The question was whether the fifth postulate has to be accepted as an independent assumption or could be deduced from other postulates. Many proofs of the fifth postulate produced during the centuries invariably turned out to fail. Around 1830, three mathematicians—Nikolai Ivanovich Lobachevsky (1793–1856), Janos Bólyay (1802–1860), and Carl Friedrich Gauss (1777–1855)—demonstrated independently but almost simultaneously that one can obtain a new geometry, a geometry that is absolutely consistent from a logical point of view, based on the negation of Euclid’s fifth postulate. This shows that Euclid was right: The fifth postulate is an independent assumption and cannot be derived from other postulates. This long expected conclusion was overshadowed, however, by the fact that a new non-Euclidean geometry was possible.
Soon it became manifest that by playing with axioms an infinite number of geometries could be created. In fact, in the second half of the nineteenth century many new geometric systems were created and extensively explored. The philosophical significance of this mathematical revolution was comparable to that of Copernicus (1473–1543): Humans are not only creatures from the outskirts of the universe, but even the universe, at least conceptually, is not unique; it is a member of an infinite family of geometric universes.
German mathematician Georg Friedrich Bernhard Riemann (1826–1866) in his 1854 inaugural lecture created a broad conceptual setting for modern geometry, which admitted more than three spatial dimensions. He also foresaw its physical applications: The world, with all its physical fields, could be but a system of fluctuating geometries.
