Note the use of the word “geometries” in the title of the textbook. For our purposes, a “geometry” is a theory or “system” of geometry. By this I mean a set of axioms, definitions, and undefined terms together with the set of all theorems that one can prove from these. The most familiar example of a geometry is Euclidean geometry. This is the geometry that we normally study in high school. In this course we will study Euclidean geometry as well as other geometries. For each geometry that we study, we will begin with a set of axioms, definitions, and undefined terms. We will then proceed to prove as many theorems as possible. In portions of Chapter 2, we will the study the very idea of a proof.
Our axiomatic approach to geometry is motivated by the geometry of the ancient Greeks. The ancient Greeks may have been the first people to develop mathematical theorems by means of logical reasoning and mathematical proof. This is perhaps one of the reasons why many modern scholars hold the ancient Greeks in very high regard. In this course we will imitate and improve upon the type of logic and reasoning that the ancient Greeks used. This type of activity is a good intellectual exercise which will help to prepare you for further study of advanced mathematics. It will be particularly helpful for those of you who plan to teach high school geometry.
Much of the course will deal with the so-called Euclidean “parallel postulate.” If we omit the parallel postulate from our accepted set of axioms, we obtain a non-Euclidean geometry. One motivation for doing this is the fact that many people over the course of history have questioned the validity of the parallel postulate. They have questioned whether or not it reflects the physical reality of the world in which we live. Some of you may feel that this type of skepticism is preposterous. In fact, it is not as preposterous as you might think. In any case, we may use non-Euclidean geometries to study the way in which the parallel postulate affects the types of theorems that we can logically prove from our set of axioms. The focus will be on determining which theorems we can prove from our axioms. Whether or not these theorems reflect the reality of our world is of less interest.
The specific course objectives are as follows:
We will cover as much of the first six chapters of the text as possible. If time permits, we may also cover portions of Chapter 7.
1:00 P.M.-3:00 P.M., Wednesday, December 14, 2005.