9_2+-+Undefined+terms,+axioms,+and+beginning+proofs+(oh+my!)

At the beginning of class, Brett explained the way the group at his table (including me) addressed the school boundaries problem. We looked at individual points to determine which school they should attend. We revisited the idea of drawing boundary lines from the midpoints of each line segment that connected the schools. This left us with the unincorporated area at the top whose children would have to be bussed elsewhere (or take online classes!). The debate resulted in re-evaluating what a "straight" line was in Ideal City and discussing how we could break down the problem into smaller problems and then look for a pattern. Many of us thought we saw a pattern, using lines from the midpoints that we had discussed earlier. A problem arose when one of the points didn't fit that pattern. However, a new pattern emerged-a curved line. We were then able to find the boundaries for Roosevelt and Franklin and our homework was to find the boundary for Jefferson.

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Continuing with the "straight line" debate, we went back to the paper, can and ball assignment. Some ideas from our assignment were discussed, including using a ruler, tracing the lines on lined paper, or snapping a chalk line, which is a chalky string anchored at two endpoints which is then snapped to create a straight line. The question was posed "How do you check if something is straight?" Could you use a chalk line on a can or a sphere? Is it possible to draw a straight line on a sphere? These remain unanswered but something to consider.

After all of the discussion, we were introduced to the idea that a line is one of Hilbert's undefined terms and we still haven't defined "straight" definitively. Hilbert suggested other undefined terms including point, lies on, between, and congruent. Just because these terms are undefined doesn't mean they don't exist. They exist, but only in relationship to each other. This led us to the definition of axiom, which is a truth that does not need to be proved. Axioms are the building blocks we use to prove theorems.

We discussed Poincare's statement that "One geometry cannot be more true than another: it can only be more convenient." The idea is that there is more than one way to represent what we see in the physical world and that one way is not more correct than another. For example, there are many ways to represent points. We can show them as {A, B, C}, or (x,y) where x,y are elements of the Real numbers, or we can draw three points on the board. These are different representations, but one is not more correct than the other.

We were introduced to three axioms that describe the relationships between points and lines. They are as follows:

I-1: For any two distinct points there exists a unique line tha tpasses through both of them. I-2: For any line there exist at least two distinct points incident with it. I-3: There exist three distinct points with the property that no line is incident with all three of them.

These are the basis for proving subsequent propositions, which is another homework assignment.

We ended with adding the definition of "parallel lines" to our list. Two lines are parallel if they have no points in common (no point lies on both of them).

Comments: ~We also proved Proposition 1.1. This proposition states "There is a line". We proved it by using I-3 to state that "there exist three distinct points with the property that no line is incident with all three of them". We chose points A, B, and C. If we pick two of these points, say A & C, we know that there is a unique line through A and C by I-1, which states "For any two distinct points **there exists a unique line** that passes through both of them". Therefore we know that a line exists.

Great job! This is a very thorough account of what happened for this class!