As A Description of The Structure of Space
Euclid believed that his axioms were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms, in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called Euclidean motions, which include translations and rotations of figures. Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries (in other words, space is homogeneous and unbounded); postulate 4 (equality of right angles) says that space is isotropic and figures may be moved to any location while maintaining congruence; and postulate 5 (the parallel postulate) that space is flat (has no intrinsic curvature).
As discussed in more detail below, Einstein's theory of relativity significantly modifies this view.
The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite (see below) and what its topology is. Modern, more rigorous reformulations of the system typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space (as in elliptic geometry), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry).
Read more about this topic: Euclidean Geometry
Famous quotes containing the words description, structure and/or space:
“The great object in life is Sensation—to feel that we exist, even though in pain; it is this “craving void” which drives us to gaming, to battle, to travel, to intemperate but keenly felt pursuits of every description whose principal attraction is the agitation inseparable from their accomplishment.”
—George Gordon Noel Byron (1788–1824)
“Agnosticism is a perfectly respectable and tenable philosophical position; it is not dogmatic and makes no pronouncements about the ultimate truths of the universe. It remains open to evidence and persuasion; lacking faith, it nevertheless does not deride faith. Atheism, on the other hand, is as unyielding and dogmatic about religious belief as true believers are about heathens. It tries to use reason to demolish a structure that is not built upon reason.”
—Sydney J. Harris (1917–1986)
“Mere human beings can’t afford to be fanatical about anything.... Not even about justice or loyalty. The fanatic for justice ends by murdering a million helpless people to clear a space for his law-courts. If we are to survive on this planet, there must be compromises.”
—Storm Jameson (1891–1986)