Geometrical Theories
A well-known example was the development of analytic geometry, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curves and surfaces of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques. That is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.
At the end of the 19th century, Felix Klein noted that the many branches of geometry which had been developed during that century (affine geometry, projective geometry, hyperbolic geometry, etc.) could all be treated in a uniform way. He did this by considering the groups under which the objects were invariant. This unification of geometry goes by the name of the Erlangen programme.
Read more about this topic: Unifying Theories In Mathematics
Famous quotes containing the word theories:
“Whatever practical people may say, this world is, after all, absolutely governed by ideas, and very often by the wildest and most hypothetical ideas. It is a matter of the very greatest importance that our theories of things that seem a long way apart from our daily lives, should be as far as possible true, and as far as possible removed from error.”
—Thomas Henry Huxley (1825–95)