2.1 - 2.3 - 2.1 An Introduction to Axiomatics and Proof 2.2...

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2.1 An Introduction to Axiomatics and Proof 2.2 The Role of Examples and Models 2.3 Incidence Axioms for Geometry Homework assignment 1 1
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2.1 An Introduction to Axiomatics and Proof An axiomatic system is a formalized construct that is used in business, religion, and mathematics – to name a few fields – and by many people, unconsciously. It provides a way to organize what is known (or believed to be “the way things are”) and to make assertions and predictions about why things happen or what’s an inevitable consequence of a happening. In an axiomatic system, once the axioms are accepted, all that follows is internally logical. Often, in real life, the big problem is understanding just what the axioms are. Luckily in this course, we will stick to math structures and handle nothing personal or controversial. And our axioms will be spelled out in writing right up front. Here’s the structure: 0 th level: Underlying foundations of arithmetic and logic (See the Primer for these, especially pages 2 and 3.) 1 st level: Undefined terms A brief list of nouns and the occasional adverb or verb. Some examples: point, line, on point, line, space, intersect You may visualize these nouns as objects and the spatial relationships (called incidence relations) as something physical and quite usual or get very creative. 2 nd level: Axioms A set of rules. Generally, the higher the level of math, the briefer the set. An eighth grade geometry text may have 30 axioms for Euclidean Geometry; a graduate textbook might have 6 for the same geometry. Axioms are always true and cannot be contradicted if you’re working in the geometry they describe. 3 rd level: Definitions Terms that can be defined by using axioms, the undefined terms, other definitions, and theorems. 4 th level: Theorems A statement about properties of the geometry or objects in the geometry that can be shown to be true using a logically developed argument called a proof. 2
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There are several examples of small axiomatic systems in the text. I have added in the structural elements that the author didn’t show in sections 2.1 and 2.2 so you can see the larger context. I have also included some additional discussion and examples for you to ponder. These additions are part of the course and may appear on tests. Text example 1 p. 52: The Three Axiom Geometry* Undefined terms: point, line, contained, intersect Axioms: 1. Each line is a set of four points. 2. Each point is contained by precisely two lines. 3. Two distinct lines that intersect do so in exactly one point. Definition: Parallel lines are lines that share no points. * I’ve just given it a name and some structure to clarify the discussion coming up. Question: Do parallel lines exist in this geometry? We will illustrate the answer without proving it. This figure looks a bit like the first iteration in the book.
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