1.1 The Structure of Geometry: Definition, Postulate, Theorem Wherein we meet for the first time three fundamental notions - definition, postulate and theorem.

What's Most Important If I were to ask you what geometry is, I'd guess that you'd say something about lines, angles, shapes and the like. I don't deny that those sorts of thing are important in geometry. They are its subject matter. But they’re not what's most important. What's most important in geometry is not its subject matter. What's most important is its method.

How to Play Chess I want to build an analogy, an analogy between chess and geometry. It will help to explain what I mean by method. To play chess, you must know two things: (a) what the pieces are (this is a rook and it's placed back row corner, etc.), and (b) the rules that govern how the pieces are moved (rooks move along any unobstructed row or column, etc.). Once you've learned these two things, you can play a game. (Perhaps not well, but you can play.)

Chess and Geometry Just like in chess, you must know two things to "play" the game of geometry. You must know what the "pieces" are; you must know how they "move". What does this mean? The pieces in geometry are the objects of geometry - points, lines, planes, angles and all the rest. To know what they are is to know their definitions. The rules in geometry are the postulates. They tell us what can be done with the objects.

How to "Play" Geometry The analogy is almost complete. But we still need to say how we "play a game" of geometry. In chess, once we know the pieces and the rules, we can play a game. In geometry, once we know the definitions and the postulates, we can prove theorems. That's really the heart of geometry. To do geometry, to really do it (and indeed to really do any sort of mathematics) is to prove theorems.

Definition, Postulate, Theorem Geometry has a certain structure - always has, always will. We first define our terms. Next we state our postulates. Finally we prove our theorems. (Actually we tend to mix the three. We give a few definitions and postulates, prove a few theorems, then add to the definitions and postulates, prove some more theorems, etc.) Remember this: every statement in geometry is a definition, a postulate or a theorem.

A Little Example I don't expect you to really understand all this yet (except perhaps for the bit about definition - I expect that you know what a definition is). So let's have a little example to illustrate. Below is a very simple mathematical system. It begins with a set of definitions and postulates. Once they have been set out, we'll prove a theorem or two.

The Definitions A theven is a positive integer divisible by 3. A thodd is a positive integer that leaves a remainder of 1 when divided by 3. A thodder is a positive integer that leaves a remainder of 2 when divided by 3.

Examples Classify as theven, thodd, or thodder: 1.1 is a ____________. 2.2 is a ____________. 3.3 is a ____________ is a ____________ is a ____________.

Postulates 1.A theven plus a theven is a theven, a theven plus a thodd is a thodd, and a theven plus a thodder is a thodder. 2.A thodd plus a thodd is a thodder, and a thodd plus a thodder is a theven. 3. A thodder plus a thodder is a thodd.

Let Us Prove Let us prove that a thodd plus twice a thodder is a thodder. Here’s how we’ll do it. We’ll ask what we get when we add a thodder to a thodder. Whatever that is, we’ll take it and ask what we get when we add it to a thodd. All answers will be provided by the postulates.

The Proof

Proofs are Complete Explanations Notice that our proof is simply an explanation of why the claim that a thodd plus twice a thodder is a thodder is true. It was moreover a complete explanation. It has not gaps. It made no jumps. You should now be convinced that a thodd plus twice a thodder is a thodder. You should understand completely why it’s true.

Summary We began with definitions of the ‘theven’, ‘thodd’ and ‘thodder’. We gave a set of postulates that described relations among the terms defined. We proved a new relation among those terms. This is the basic structure of all of mathematics. We define terms, state postulates and prove theorems.