Section 3-3: Justifying Conclusions (Day 19)

Section 3-3 of the U of Chicago text is an introduction to proof. Because the Common Core Standards specifically mention statements that students are supposed to prove, that makes this section one of the most important sections of the book -- and that's why I cover this section before skipping to Chapter 4.

But what exactly is a proof? The following definition of proof comes from a professional mathematician:

A proof of a statement Phi consists of a finite sequence of statements, each of which is either an axiom, or follows from previous statements by logical inference such that Phi is the last statement in the sequence.

Notice that this is not that much different from the definition given in the U of Chicago text. Of course, a proof must be a finite sequence of statements -- proofs can't go on forever! The U of Chicago states that valid justifications in proofs include postulates (note that "axiom" is basically another word for "postulate") and theorems already proved (the "previous statements" mentioned above). But what about the third justification mentioned in the text -- definitions? Strictly speaking, a definition is also a special type of axiom, called a "definitional axiom." And of course, the last statement in the sequence is "Phi" -- that is, the statement that we're trying to prove!

Now Section 3-2 of the text mentions two theorems, the Linear Pair and Vertical Angle Theorems. But I left these out, since they didn't fit on my Frayer model page from yesterday. But those theorems certainly fit here in 3-3, for after all, the first example of a proof in the text is that of the Vertical Angle Theorem.

The proof of the Vertical Angle Theorem in the text is sort of a hybrid between a paragraph proof and a two-column proof. The conclusions and justifications aren't written in two-column form, but since each conclusion is followed by its justification, it might as well be a two-column proof. Subsequent proofs in this section are essentially paragraph proofs -- actual two-column proofs don't appear until Section 4-4.

One thing I like about this section is that it gives the reasons why anyone would want to write a proof. The first one is:

"What is obvious to one person may not be obvious to another person. Sometimes people disagree."

As I mentioned before, when asked what the least favorite part of geometry class is, a very common answer is, proofs. But this is what professional mathematicians do all day -- a common joke is that a mathematician is a machine for turning coffee into theorems. (This line is usually attributed to the 20th-century Hungarian mathematician Paul Erdős.) Recall that a theorem is a statement that has been proved -- so Paul is telling us that mathematicians are machines that prove things.

Indeed, some of the most famous math problems in the world are proofs. About 400 years ago, a French mathematician named Pierre de Fermat (actually he was a lawyer -- but then again, both lawyers and mathematicians are known for proving things) made a very innocent-looking statement:

"No three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two."

But Fermat was unable to write a proof of this statement -- at least, not a proof that he could fit in the margin of the book he was reading. It was not until 20 years ago when a British mathematician named Andrew Wiles finally proved of Fermat's Last Theorem. His proof is extremely complicated -- no wonder it took over 350 years for anyone to prove it!

Even today there are statements that appear to be true, but no one has proved them yet. The Clay Mathematics Institute has offered a prize of one million dollars to the first person who can prove each of the seven Millennium Problems (so called because the prize was first offered at the start of this millennium). So far, only one of the problems has been proved, so six million dollars remain unclaimed.

And we can go from problems that take years -- or even centuries -- to prove, to some which take a few hours to solve. Every year on the first Saturday in December, college students from around the country participate in the Putnam competition. There are twelve questions -- most or all of which are proofs -- and six hours in which to solve them. And if you can get even one of the twelve questions correct, then you will have one of the top scores in the country!

Now let's compare this to the attitude of many high school geometry students -- mathematicians may spend hours, years, even centuries to write a proof, yet the students can't spend a few minutes proving the Vertical Angle Theorem?

There's a wide range of beliefs on how much proof there should be in a geometry course -- from David Joyce, who believes that anything that can be proved should be proved as soon as possible, all the way to Michael Serra, who doesn't prove anything in his text until Chapter 14. On this blog, I'll take Joyce's approach, but only for proofs emphasized by the Common Core.