> It is eminently reasonable to believe that the purpose of mathematics is > to provide a rigorous and practically useful conceptual framework for > reasoning quantitatively about real world phenomena. The objects that > exist in a conceptual framework are concepts; the universe of > mathematical objects is a collection of concepts. Concepts can be > encoded in language. Languages are countable. The Cantorian claim that > "uncountable" infinite collections exist is tantamount to the claim that > mathematics should assert the existence of things that are not within the > mathematical conceptual framework.
Euclidean geometry requires actually infinite sets of points, lines, rays and line segments, circles, etc. and an actual infiniteness in the at least one definition of parallelism.
And non-Eucliedan geometry gives us the opportunity to have an actually infinite number of geometries. --