G 6
SOME HISTORY*
In the beginning, geometry (which means earth-measure) involved measurement to a great extent and was actually a collection of procedures arrived at through experimentation, observation, guessing, and occasional flashes of intuition.
Examples:
Around 600 B.C., Thales introduced abstraction into geometry. He conceived geometric figures as abstract shapes and noticed that some geometric facts (theorems?) were deducible from others, and thus suggested that geometry should become a purely mental activity.
Thales' thinking was picked up by Pythagoras (c.570-c.495) and his "Society of Pythagoreans". The society accepted Thales' belief that geometry be made a deductive science. Their greatest contribution was a discovery that helped set the standard of proof. Thales had deduced his theorems by a combination of logic and intuition. The Pythagoreans discovered that logic and intuition can disagree. As an example of this, on an intuitive level, they were sure that "2was a rational number; but by logic and computation, they were equally sure that "2 was not a rational number. As a result, the mathematical world had to make a choice; and the choice made was in favor of logic. Thus, "rigor" was introduced into mathematics, and geometry became viewed as a formal axiomatic system. It should be noted that intuition in mathematics has not been banished. Any formal branch of mathematics builds on a set of axioms (statements accepted without proof primarily because of their intuitive appeal). However, intuitive evidence is not accepted as conclusive.
*Sources:
Smart, J. Modern Geometries. 3rd Edition. Belmont, CA: Brooks/Cole, 1968.
Trudeau, R. The Non-Euclidean Revolution. Boston, MA: Birkhauser, 1987.