8/9/2019 Lecture 5_Notes Axiomatic System Discussion

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Geometry Dr. Perdue

Notes from Axiomatic System Discussion

Geometry, like many branches of mathematics, did not simply fall from the sky in its

present, completed form. Instead, mathematicians of the past made it up! They were able to make

up everything we now know to be geometry based on certain rules. In essence, geometry is based

upon a set of rules that govern an axiomatic system. The first, and most famous, mathematician

to make up a geometry axiomatic system was Euclid. However, he was not the only one! Two others,

Riemann & Lobachevsky, also created geometry systems. Unfortunately, their systems are rarely

taught in high school and most people are ignorant of their very existence. Axiomatic systems

consist of 4 parts: undefined terms, definitions, axioms (also known as postulates), and theorems.

Undefined terms are just that, undefined. They are words that can be used based on an

intuitive understanding of them (in Euclids system, he used three of them: point, line, & plane).

These terms are given, we do not have to define them, prove them, or do anything else to use them.

Definitions are given to introduce new terms. Definitions must use terms that are

previously defined or undefined. Example #1, once Euclid has point as an undefined term, he can

DEFINE the term between to describe a point in relation to two other points. Example #2, once

Euclid has point and line as undefined terms and between as a defined term, he could then

DEFINE line segment in terms of those terms (a line segment is the portion of a line that is

between two given points).

Axioms or postulates are the third part to an axiomatic system. These are statements

about the undefined & defined terms that are assumed to be true. They are just given; they do

not need to be proven. In Euclids system, he had 5 axioms. An example of one of them is A plane

consists of at least 3 points. Basically, he had to make this assumption because he needed two

points to make up a line and he needed a point not on the line to get things like perpendicular lines.

Another example was Euclids 5th postulate, also known as his parallel postulate. This was the most

controversial of all because it claimed the existence and uniqueness of parallel lines. It was stated

like this: Given a line and a point not on the line, there is one and only one line that is parallel to

the given line and passes through the given point. The interesting thing is that those other two

mathematicians mentioned earlier, Riemann and Lobachevsky, used all the same parts of Euclids

axiomatic system in their systems with the exception of Euclids 5th postulate. Basically, where

Euclid claimed existence & uniqueness, one of them (Riemann) claimed they (parallel lines) didnt

exist at all and the other (Lobachevsky) claimed they did exist but were not unique (in other words,

there were more than one line parallel to a given line passing through a given point). See the

exploration mentioned in Chapter 13 of your text for more details about this.

The last part to an axiomatic system is the theorems. Theorems are statements about the

undefined & defined terms that must be proven true. The way they are proven is by using the four

methods of logical reasoning weve learned (M.P., M.T., L.C. and L.S.) or will learn (if were not that

far yet in class). The interesting thing about the three types of geometries is that because of

that different axiom (assumption) that each of them makes, certain things that we believe are

true all the time are no longer. For example, its only in Euclids geometry that the sum of the

interior angles of a triangle is exactly equal to 180 degrees. In the other two, that sum is either

always greater than or always less than 180 degrees. Many other theorems are drastically

different as well.