Euclid's Axiomatic Approach

We will introduce seven undefined terms in all as we develop geometry: point, line (this means a straight line, but we will not repeat the word "straight" every time), plane, ruler, distance, area, and volume. We will also use the word space to mean the set of all points. Everything else will be defined in terms of these. In this section we will deal with the axioms which just involve the first three of these terms (point, line, and plane), and which do not involve any quantitative measurement.

Classical geometry provides a rich vocabulary to describe the various possible relationships between points, lines, and planes. For example, a point $P$ may lie on the line $\mathcal{L}$, or the line $\mathcal{L}$ may pass through or meet or contain the point $P$. These all mean the same thing, which in the notation of set theory is written $P \in \mathcal{L}$. Similarly a line may lie in a plane, two lines may meet at a point, and so on. It helps to have all these different terms available, though from a severely abstract point of view they are only picturesque ways of saying whether two sets intersect or not. A catch-all term for these relationships is "incidence relations", and the axioms in this section, which deal with the properties of "incidence relations", are often known as the axioms of incidence. For each axiom, I suggest you draw some sketches and convince yourself that the axiom is true in your intuitive picture of geometry. This process should not be too hard in the case of the first of our axioms:

Axiom (Line Axiom). Through any two distinct points there is exactly one line.

If $P$ and $Q$ are points, this allows us to introduce the notation $\overleftrightarrow{PQ}$ for the unique line containing both $P$ and $Q$. The two-sided arrow is meant to remind you that the line extends as far as you like past $P$ and $Q$ in both directions. Later on, we will distinguish $\overleftrightarrow{PQ}$ from $PQ$, which will refer just to the segment of $\overleftrightarrow{PQ}$ that is between $P$ and $Q$.

Even from this single axiom we can derive a consequence which is worth noticing:

Proposition. Two distinct lines can meet in at most one point.

Proof. If they met at two points, say $P_1$ and $P_2$, then there would be at least two lines through $P_1$ and $P_2$, contrary to the line axioms.$\,$ $\square$

Of course, it is perfectly possible for two lines not to meet at all. In intuitive terms they can either be parallel (in the same direction but never meeting) or skew (in different directions but not in the same plane). We will have to discuss parallel lines carefully later on.

Before we introduce any further axioms it may be helpful to explain a modern way of thinking about axiomatic systems in general. This way of thinking emphasizes the fact that the basic terms of the axiomatic system are undefined. They could mean anything! Until we know what they mean, there is no point in asking whether or not the axioms are true. Suppose that alien beings have landed on Earth by flying saucer, and their leader tells you that:$$\text{Through any two distinct blurgs there is exactly one phogon.}$$Unless you know what a blurg and a phogon are, you will have no way of telling whether or not this statement is true. In the same way, there may be many different interpretations of the undefined terms point and line in an axiomatic system for geometry. Some interpretations will make all the axioms true and some will not. An interpretation which makes all the axioms true is called a model for the axiomatic system; because the theorems are all deduced logically from the axioms, they will be true in a model as well.

Example. Here are two different models for the single axiom that we have introduced so far. They are much smaller than the usual model we have in mind for geometry; in fact, Model 1 has just four points and Model 2 has seven. Because they have only finitely many points, we can describe them just by listing all the points and lines, as follows:

Model 1 — Points: $A$, $B$, $C$, $D$; Lines: $AB$, $AC$, $AD$, $BC$, $BD$, $CD$

Model 2 — Points: $A$, $B$, $C$, $D$, $E$, $F$, $G$; Lines: $AFB$, $BDC$, $CEA$, $AGD$, $BGE$, $CGF$, $DEF$

The models may be easier to visualize if you draw a diagram like the following.

However, remember that only the labelled points are really part of the model. The rest of the diagram is just to help you see what is going on. It is easy to check that in both models the line axiom is satisfied.

Such models are called finite geometries, and their study is an active of mathematics in its own right, with applications to the design of scientific experiments and error-correcting codes for computers. It's dubious any part of the Geometry part of Expii will touch upon it more, though.

(In progress...)