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 is of course now standard too in physics.
When topology is routinely described in terms of properties 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 groups – 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 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 structure."
For a geometry and its group, an element of the group is sometimes called a motion of the geometry. For example, one can learn about the Poincaré half-plane model of hyperbolic geometry through a development based on hyperbolic motions. Such a development enables one to methodically prove the ultraparallel theorem by successive motions.
The Erlangen Program is carried into mathematical logic by Alfred Tarski in his analysis of propositional truth.
Read more about this topic: Erlangen Program
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