Tuesday, August 12, 2014

Section 1-7: Postulates (Day 4)

Section 1-7 of the U of Chicago text introduces postulates. In the last section, the undefined terms -- the primitive notions -- pointline, and plane were introduced. Since these are undefined, we don't really know what they are unless we have postulates -- also known as axioms -- to describe them.

I reproduce the main postulate of this section, the Point-Line-Plane Postulate:

Point-Line-Plane Postulate:

(a) Unique line assumption: Through any two points, there is exactly one line.

(b) Dimension assumption: Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.

(c) Number line assumption: Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.

(d) Distance assumption: On a number line, there is a unique distance between two points. If the points have coordinates x and y, we define this distance to be |x - y|.

Let's look at each of these four assumptions -- since postulates really are assumptions, or statements that are obviously true -- in detail. The first assumption, that two points determine a line, goes all the way back to Euclid's First Postulate. In Hilbert's formulation of Euclidean geometry, this is Axioms I.1 and I.2.

The second assumption, about dimensions, are often different in other texts. Some texts, for example, emphasize that three noncollinear points determine a plane -- and give the example of a tripod standing on its three legs, the ends of which are the three points determining the plane of the floor. Hilbert's Axioms I.3 through I.8 roughly correspond to this assumption.

Assumptions (c) and (d) often appear in geometry textbooks as the "Ruler Postulate." The Ruler Postulate was first formulated by the American mathematician George David Birkhoff, about eighty years ago. The Ruler Postulate basically states that rulers work -- that is, we can measure line segments.

The section continues with its first theorem, the Line Intersection Theorem:

Line Intersection Theorem:
Two different lines intersect in at most one point.

And then we have the definition of parallel lines:

Two coplanar lines are parallel lines if and only if they have no points in common, or they are identical.

I've discussed these in some of my introductory posts in July. The last four words of this definition are controversial: "or they are identical." But as I pointed out, using this definition often simplifies later proofs -- in particular, it often allows one to replace an indirect proof with a direct proof. And technically speaking, the proof of the Line Intersection Theorem is actually an indirect proof -- but it's so simple that the text includes an informal argument here while delaying other indirect proofs until Chapter 13.

So far, in the introductory posts, we wanted to prove that two (coplanar) lines are parallel -- using our definition that they are either non-intersecting or identical. We were able to do this two ways -- we could prove that if they have at least one point in common, then they must have every point in common -- or we could prove that if at least one point on one line fails to lie on the second line, then every point on the first line fails to lie on the second line. These are hypotheses and conclusions that can easily fit into the Given and Prove sections of a proof.

But as teachers, our priority is to make geometry easy for the students to understand. So which will confuse the students less: a definition of parallel containing those four extra words "or they are identical," or many indirect proofs? We can't be sure until actually teaching this in a classroom.

So for now, I will stick to the U of Chicago definition of parallel, with those words "or they are identical," and delay indirect proofs as long as possible. But on the following images, I'll just leave a space for "parallel" in the vocabulary section and leave it up to individual teachers whether or not to include those four extra words in the definition. (Notice that in my exercises derived from the U of Chicago text, I preserved the true or false question "a line is parallel to itself." Of course, the answer will depend on which definition the teacher decides to use.)

The section concludes with some Postulates from Arithmetic and Algebra. As I mentioned yesterday, I want to avoid mentioning "algebra" -- the subject that causes many students to hate math -- yet these are important properties that show up in proofs (for example, the Reflexive Property of Equality). And so I'll just call them "Properties from Arithmetic" and just leave "algebra" out of it.

As an aside, let me point out that I was once a teenager studying geometry. Like those of many students at that age, my thoughts turned toward girlfriends and boyfriends -- except that I was the geometry nerd who wasn't a part of all of that. Like most nerds, I could make the following assumptions:

David Walker's First Postulate:
I am physically attracted to no other person.

David Walker's Second Postulate:
I am physically attractive to no other person.

These are statements that were obviously true -- in other words, they're postulates.

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