Lesson 13-2: Negations (Day 132)

This is what Theoni Pappas writes on page 95 of her Magic of Mathematics:

"Changes can be made from an oil painting effect to a watercolor, and brush shapes can be changed at whim. Minute areas can be magnified and reworked more easily."

Here Pappas is explaining why an artist would prefer a computer to traditional media. She again mentions Leonardo da Vinci and assures us that the Italian would have embraced computer art had it existed in his day. Indeed, da Vinci was mathematically as well as artistically inclined, and he is said to have used the famous golden rectangle in creating some of his most famous masterpieces, including The Last Supper (and that's appropriate, considering that Easter is around the corner).

I don't have much to say from last year about today's Lesson 13-2, since I combined it with 13-1 and another lesson last year. Here is what little I did say on negations, followed by a single worksheet:

For example, the statement:

All unicorns are white.

is actually true -- after all, we have never seen a unicorn that isn't white (precisely because there exists no unicorns at all, much less ones that aren't white). Another way of thinking about this is that there are zero unicorns in this world, and all zero of them are white! In if-then form this statement becomes:

If an animal is a unicorn, then it is white.

The hypothesis is false (since there are no unicorns), so the entire conditional is true. This statement has no counterexamples (unicorns that aren't white), and conditionals without counterexamples are normally called true.

The book then derives, from the statement 1=2, the statement 131=177. There is a famous example of a derivation of a false conclusion from a false hypothesis, often attributed to the British mathematician Bertrand Russell, about a hundred years ago. From the statement 1=2, Russell proved that he was the Pope:

The Pope and I are two, therefore the Pope and I are one.

that is, he used the the Substitution Property of Equality from the hypothesis 1=2.

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.

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

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