Question

Can someone write a 2 page paper on Teaching Philosophy regarding geometry . ?

Answer 1

Epistemology of Geometry

Geometrical knowledge typically concerns two kinds of things: theoretical or abstract knowledge contained in the definitions, theorems, and proofs in a system of geometry; and some knowledge of the external world, such as is expressed in terms taken from a system of geometry. The nature of the relation between the abstract geometry and its practical expression has also to be considered.

This essay considers various theories of geometry, their grounds for intelligibility, for validity, and for physical interpretability in the period largely before the advent of the theories of special and general relativity in the 20th century. It turns out that a complicated interplay between shortest and straightest is at work in many stages.

Before the 19th century only one geometry was studied in any depth or thought to be an accurate or correct description of physical space, and that was Euclidean geometry. The 19th century itself saw a profusion of new geometries, of which the most important were projective geometry and non-Euclidean or hyperbolic geometry. Projective geometry can be thought of as a deepening of the non-metrical and formal sides of Euclidean geometry; non-Euclidean geometry as a challenge to its metrical aspects and implications. By the opening years of the 20th century a variety of Riemannian differential geometries had been proposed, which made rigorous sense of non-Euclidean geometry. There were also significant advances in the domain of abstract geometries, such as those proposed by David Hilbert. It follows that the terms ‘geometry’ and ‘physical space’ do not have simple meanings in the 19th century, and changing conceptions of these terms do not follow a simple pattern of refinement. Their inter-relations therefore also have a complicated history.