Synthetic geometry
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Synthetic geometry is the branch of geometry which makes use of theorems and synthetic observations to draw conclusions, as opposed to analytic geometry which uses algebra to perform geometric computations and solve problems.
The geometry of Euclid was indeed synthetic, though not all of the books covered topics of pure geometry. The heyday of synthetic geometry can be considered to have been the 19th century; when methods based on coordinates and calculus were ignored by some geometers such as Jakob Steiner, in favour of a synthetic development of projective geometry.
For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory (with more models) than is found by starting with a vector space of dimension three. The close axiomatic study of Euclidean geometry led to the discovery of non-Euclidean geometry.
Another example concerns inversive geometry as advanced by Ludwig Immanuel Magnus , which can be considered synthetic in spirit. The closely related reciprocation expresses analysis of the plane. As for hyperbolic geometry, it was initially advanced by Lobachevsky and Bolyai in a synthetic spirit, but it took the analytic models made using Mobius transformations to make it believable. On the other hand, the theory of special relativity developed analytically with the linear algebra of Lorentz transformation; more confidence in the foundation of spacetime theory may evolve from the synthetic approach given by Lewis and Wilson in 1912 (see reference "Synthetic Spacetime").
If the axiom set is not categorical (so that there is more than one model) one has the geometry/geometries debate to settle. That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now starting point for theory rather than self-evident plank in platform based on intuition. And since the Erlangen program of Klein the geometrical nature of a geometry has been seen as the connection of symmetry and the content of propositions, rather than the style of development.
In relation with computational geometry, a computational synthetic geometry has been founded, having close connection, for example, with matroid theory. Synthetic differential geometry is an application of topos theory to the foundations of differentiable manifold theory.
