We’ve been talking about axiom systems and formal proofs in mathematics. On Wednesday, we covered some things from axiomatic Euclidean geometry. This is formal and a bit abstract. The idea is that we’ll talk about axioms of fair voting systems on Friday and show that certain voting systems fail to satisfy those axioms.

Undefined Terms

We have to start somewhere, and so we’re going to agree on what the following terms mean without defining them.

  • point
  • line
  • lies on (as in, a point lies on a line)
  • between (as in, a point is between two other points on a line)
  • circle
  • congruent

The last one there is a bit of a mathematical term. Congruence is a bit weaker than equality, just as equality is a bit weaker than identity. I explained this in class with some examples. Shoot me an email if you have questions or would like more examples.

Axioms

We’re going to be using the following axioms for Euclidean geometry.

  1. We may draw a (unique) line connecting any two points.
  2. We may extend lines indefinitely in either direction once they are created.
  3. Given a point and a length, we may create a circle centered at the point with radius equal to the length.
  4. All right angles are congruent.
  5. Given a line and a point not on the line, there is a unique line going through the point that is parallel to the given line.

Defining a Right Angle

We didn’t include “right angle” in the list of undefined terms. Should we? Well, if we can construct the definition using what we have so far, then no. Can we describe what a right angle is using only what we have? The answer is yes. But, it isn’t easy. We need to define some things along the way. First, we need to know what a “ray” is.

Definition: Given two points $A$ and $B$, the ray $AB\rightarrow$ is the line starting at $A$, containing $B$, and then continuing indefinitely past $B$.

Once we have the definition of a ray, we can then define an angle.

Definition: Given three points $A$, $B$, and $C$, the angle $\angle BAC$ is the ray $AB\rightarrow$ together with the ray $AC\rightarrow$. That is, both rays start at $A$, one containing $B$ and one containing $C$.

Definition: Two rays are called opposite if they start at the same point, lie on the same line, but point in opposite directions.

Definition: Two angles are called supplementary if two of the rays forming them are opposite.

In the diagram, angles 1 and 2 are supplementary, and the rays from $B$ to $A$ and $C$ are opposite. Now, we can define right angles.

Definition: An angle is a right angle if there exists a supplementary opposite angle to it.

Proofs

Sometimes, we can prove things directly from the axioms, and other times we can’t. When we can’t, we use formal logic to guide us through other strategies. One possible approach is what we’ll can RAA (reductio ad absurdum). The idea is that we want to show that a collection of known facts, axioms, definitions and so on can be used to prove a claim. So, what we do is assume that the claim is false … and then show that doing this results in running up against a logical brick wall. I explained this in detail in class, and you should feel free to ask me questions about it. It isn’t easy, so don’t think that I’m expecting you to get it immediately. The example we used in class was to prove that $\sqrt{2}$ is irrational. We assumed that is was in fact rational, and we ended up getting something that was nonsensical… meaning that one of our assumptions didn’t make sense.

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