8 - Geometry

Summary

Basic Objects: Points and Lines in Planes

Basic Goal: Axioms for Different Geometries

The axiomatic geometry of Euclid was the model for correct reasoning from at least as early as 300 BC to the mid 1800s. Here was a system of thought that started with basic definitions and axioms and then proceeded to prove theorem after theorem about geometry, all done without any empirical input. It was believed that Euclidean geometry correctly described the space that we live in. Pure thought seemingly told us about the physical world, which is a heady idea for mathematicians. But by the early 1800s, non-Euclidean geometries had been discovered, culminating in the early 1900s in the special and general theory of relativity, by which time it became clear that, since there are various types of geometry, the type of geometry that describes our universe is an empirical question. Pure thought can tell us the possibilities but does not appear able to pick out the correct one. (For a popular account of this development by a fine mathematician and mathematical gadfly, see Kline's Mathematics and the Search for Knowledge [73].)

Euclid started with basic definitions and attempted to give definitions for his terms. Today, this is viewed as a false start. An axiomatic system starts with a collection of undefined terms and a collection of relations (axioms) among these undefined terms. We can then prove theorems based on these axioms.