Sections 13-1 and 13-2: Logic and Negations (Day 104)

According to my plan, the next chapter in the U of Chicago text for us to cover is Chapter 13, which is on Logic and Indirect Reasoning. Here's the daily plan for this chapter:

Today, February 2nd: Sections 13-1 and 13-2 (Negations)
Tomorrow, February 3rd: Sections 13-3 and 13-4 (Indirect Proof)
Wednesday, February 4th: Section 13-5 (Tangents to Circles and Spheres)
Thursday, February 5th: Section 13-6 (Uniqueness)
Friday, February 6th: Activity
Monday, February 9th: Lincoln's Birthday (no school, therefore no post)
Tuesday, February 10th: Section 13-7 (Exterior Angles, includes Triangle Inequality)
Wednesday, February 11th: Section 13-8 (Exterior Angles of Polygons)
Thursday, February 12th: Review for Chapter 13 Test
Friday, February 13th: Chapter 13 Test

Recall that Chapter 2 of the U of Chicago text also covers mathematical logic, and other texts combine the material from both Chapters 2 and 13 into their own respective Chapter 2. I have considered moving some of Section 13-1 up into first semester with Chapter 2. But I haven't done that yet.

Also, I included some of Section 13-7 at the end of the first semester, since the exterior angles of triangles are simple to calculate. But my goal is to prove the Triangle Inequality, mentioned in Section 1-9 as a postulate. However, it can be proved as a theorem -- and some of the key results needed for the proof of the Triangle Inequality appear in Section 13-7.

Here I combine Section 13-1 with 13-2, as well as 13-3 with 13-4. This way, we can get into the actual geometry content of Sections 13-5 through 13-8 faster.

In today's lesson, the U of Chicago text introduces the symbol not-p for the negation of p. In other texts, the notation ~p is used, but I have no reason to deviate from the U of Chicago here.

This chapter focuses on mathematical logic, which ultimately helps the students write proofs. I mentioned earlier that the Law of Detachment is often known by its Latin name, modus ponens. In fact, I pointed out that on the Metamath website -- a website full of mathematical proofs -- modus ponens is one of the most used justifications:

http://us.metamath.org/mpegif/ax-mp.html

Notice that I only mention the Metamath website for general information. This website is definitely not suitable for use in a high school math classroom. At Metamath, even a simple proof like that of 2+2=4 is very complex:

http://us.metamath.org/mpegif/2p2e4.html

 In fact, believe it or not the proof was once even more complicated because it tried to use pure set theory to prove that 2+2=4, and then later on more axioms (postulates) were added to make the proofs easier -- similar to the postulates for real numbers mentioned in Section 1-7. To repeat, the basic idea is that one makes a proof simpler by adding more axioms/postulates.

Before leaving this site, let me point out that this site gives yet a third way of writing the "not" symbol used in negations:

http://us.metamath.org/mpegif/wn.html

This is when students often ask, "Why do we have to learn proofs?" Of course, they ask because proofs are perhaps the most difficult part of a geometry course. The answer is that even though mathematical proofs may not be important per se -- but proofs are. Many fields, from law to medicine, depend on proving things. We don't want to guess that a certain person is guilty or that taking a certain medicine is effective -- we want to prove it. For centuries, the dominant way to learn how logical arguments work was to read Euclid. We already discussed MLK's birthday and the holiday we had off in his honor, so now that it's February, let's move on to the next holiday, Lincoln's birthday, and learn about how Honest Abe learned about logical arguments from Euclid:

http://the-american-catholic.com/2012/08/16/lincoln-and-euclid/

Unfortunately, the above link is a political and religious website. Well, I suppose it's impossible to avoid politics when discussing Lincoln, but the webpage is also a Catholic site.