Plane Geometry - Bruce E. Shapiro

Section 5Logic and Proof inMathematicsThis section is intentionally concise as it should be a review of Math 320.The language of mathematics is formal. Statements can be written down ina form that separates their content from their meaning in order to establishconsistency and validity. We start with a set of undefined terms that areaccepted as given without further explanation, such as point, line, plane.Usually these terms can be defined in some further reduced terms but, likea dictionary, we will eventually run into circular definitions if we attemptto continue to refine the definitions, or we end up with a statement thatdoesn’t really make much sense:A point is that of which there is no part. [Euclid, definition 1]Does this really clarify what a point is? Thus it is best to choose ourundefined terms as something that is more-or-less agreed upon.Following the undefined terms, we can define additional objects in terms ofthe undefined.We then need to state our assumptions. These are called postulates oraxioms; in Euclid’s system there are five.Next, we have a system of rules for obtaining new true statements fromour postulates. This is our logical system. We will sometimes define newsymbols as shorthands to represent parts of our logical system.The true statements are called theorems, and the sequence of steps thatjustifies the validity of the theorem is called a proof. We will write all of19