Mathematics 223 notes - ABOUT THE COURSE
The name of the course is ``Geometry'' and the subject matter is non-Euclidean and Euclidean geometry, developed from axioms. Mathematics usually consists of three parts - discovery, which is illogical, random, and emotional; organization into proofs, which demands logic and accuracy; and formulation of the proofs into words and symbols in order to communicate one's results to others. No one can teach you how to discover what is true in mathematics and why it is true, although working hard and long on substantial problems can stretch one's mental muscles and give one experience in trying to make such discoveries. The emphasis in the course will be on careful use of language and reasoning - you will be expected to read and write mathematics with accuracy, paying attention to the technical meaning of words and phrases, writing sentences that make sense, and organizing arguments which proceed logically and really do prove what is supposed to be proved. Unfortunately for the students, neither the author of your textbook nor your instructor will be perfect in his own uses of language and reasoning; nonetheless you will be held to high standards.
The role of definitions . Words in mathematics are often given specific special meanings. Mathematical definitions are merely abbreviations, whereby a word or short phrase carries a meaning which when spelled out involves many words and interrelationships. Frequently, we will use the definitions of technical terms (such as the interior of an angle) as guides in proofs (such as proving that given certain hypotheses, a certain point is in the interior of a certain angle). In fact, the only allowable things to use in a proof are the axioms, the assumptions in the particular theorem, the previously proved facts, and the spelled out meaning of the definitions of the terms in the assumptions or conclusion. <>When we apply a definition, we don't state it in our own words, we state it precisely as it has been defined in the course. <>All of this takes some getting used to. Sometimes it will seem that the author and instructor are being very fussy. That's part of the process. In introductory courses, the teacher pays attention to the ``theory'' and the students learn by repetition and memorization how to work standard types of problems and, we hope, they also gain some understanding of the structure of the subject. In this course, all of us develop the ``theory,'' i.e., the mathematics.
Now let's try to clarify how a certain terminology which you will see often this term is supposed to be employed. If we say ``let A,B, C be points,'' we are using three different letters to name certain points, but that does not mean that those points are three different objects. If we want to include that requirement, we must say so somehow, and the way we do that is with the word ``distinct.'' So ``let A,B, C be distinct points'' means that the letters A, B, C are being used to name three points no two of which are the same as one another. So ``distinct'' means ``different'' in a certain sense. But saying ``let A be a distinct point'' makes no sense. (What does ``let A be a different point'' mean?) Also, ``Let P be a point. Let Q be a distinct point'' does not make sense. You have to write either ``Let P and Q be distinct points'' or ``Let P be a point. Let Q be a point distinct from P.'' (There might be several sentences between these two.) Likewise for lines or any other mathematical objects. If the distinctness of some points is in a conclusion instead of an assumption, what must we do in order to prove that statement? For two points, we must prove that they are not the same as one another. For three or more points, we must show that no two of them are the same as one another. That might be provable all at once or it might require considering cases. (Since we'll face such a situation right at the beginning of Chapter 3, I won't give an example now.)