Geometric Stuctures: Session 2 (Comments and Loose Ends)
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
1. Mathematical systems have undefined terms. For plane geometry these are point and line. Euclid's work, as impressive as it continues to be, suffered in that he tried to define every term (perhaps because he was trying to use a "physical flat surface" as a model?). He also failed to have axioms that would allow one to "recover" what today we think of as Euclidean geometry. He did not address the issue ofs "betweeness," "continuity" or "congruence."
2. There is a tradeoff between having to invent more words for being able to give finer grained explanations and the "baggage" of learning (memorizing) lots of new words. Think about colors. If you describe a dress as "red" in a phone conversion, the person you are speaking to might conjure up something which had the hue of a tomato, a fire engine, or blood but probably not pink. There are lots of words that are used to convey specialized colors: azure, mint, and olive. If there is a classification system for quadrilaterals that gives rise to 20 types, do we need 20 words in English, French, and German, for each of these types?
(Aside: William Gladstone, the British politician and a scholar in the work of Homer, pointed out that color seemed to play almost no role in Homer's descriptions in his writing. Scientists have speculated about the role of color and color perception in early human development.)
3. Part of the power of using the axiomatic approach is that one discovers similtaneously facts about the many structures that obey the axioms when one proves theorems in the system being studied. Thus, there are many kinds of objects which obey the axioms for group theory and theorems about groups apply to them all.
4. Students invariably have trouble with models for geometric axioms which involve a finite number points and lines. It is very natural to draw the lines as "continuous" objects which connect up the points but students assume that all of the points on these "continuous lines" are actually points of the geometry, which typically they are not. Of course, one has the option of never drawing diagrams but then, presumably, geometry should be as visual a subject as possible.
It is worth noting that geometric conjectures (what might be true) can often be obtained by geometric drawings and methods but also using algebraic approaches. For proofs, algebraic approaches are often more powerful, but not always. Thus, there are some Euclidean theorems which are most transparent using synthetic arguments while there are other Euclidean theorems which are certainly much easier to prove using algebra. Also, sometimes algebra suggests geometric results to seek out.