Editing Erlangen program (section)

==Influence on later work==

The long-term effects of the Erlangen program can be seen all over pure mathematics (see tacit use at [[congruence (geometry)]], for example); and the idea of transformations and of synthesis using groups of [[symmetry (physics)|symmetry]] has become standard in [[physics]].

When [[topology]] is routinely described in terms of properties [[invariant (mathematics)|invariant]] under [[homeomorphism]], one can see the underlying idea in operation. The groups involved will be infinite-dimensional in almost all cases – and not [[Lie group]]s – but the philosophy is the same. Of course this mostly speaks to the pedagogical influence of Klein. Books such as those by [[H.S.M. Coxeter]] routinely used the Erlangen program approach to help 'place' geometries. In pedagogic terms, the program became [[transformation geometry]], a mixed blessing in the sense that it builds on stronger intuitions than the style of [[Euclid]], but is less easily converted into a [[logical system]].

In his book ''Structuralism'' (1970) [[Jean Piaget]] says, "In the eyes of contemporary structuralist mathematicians, like [[Nicolas Bourbaki|Bourbaki]], the Erlangen program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of [[mathematical structure|structure]]."

For a geometry and its group, an element of the group is sometimes called a [[motion (geometry)|motion]] of the geometry. For example, one can learn about the [[Poincaré half-plane model]] of [[hyperbolic geometry]] through a development based on [[hyperbolic motion]]s. Such a development enables one to methodically prove the [[ultraparallel theorem]] by successive motions.