A Second, Less Trivial Example

At this point, I acknowledge that perhaps this first example is a little too simple. So I then draw a triangle on the whiteboard that looks like Figure 6.1.

Figure 6.1. Triangle.

I then point out that the triangle takes on additional meaning when I complete the diagram with the addition shown in Figure 6.2.

Figure 6.2. Right triangle.

Now this triangle is unambiguously a right triangle, because the little square doohickey is a worldwide convention meaning "right angle." Furthermore, I can now label the sides of the triangle A, B, and C, as in Figure 6.3.

Figure 6.3. Labeled right triangle.

And, immediately, I can write down that

A² + B² = C².

Now this has a few very endearing properties. First, it is once again an example of a universal notation. Right angles, right triangles, and the symbols representing them are the same all over the world; someone from ancient Egypt could in principle reason about right triangles with a modern Peruvian by drawing such diagrams. What's more, once the diagram for the right triangle has been written down, the relationship of A, B, and C is defined. A, B, and C can no longer have completely arbitrary values; once any two of them are specified, the third is determined as well. The diagram implies the Pythagorean Theorem. One could even go so far as to say that the diagram has some "semantics," that there is a well understood relationship between the picture and the values implied by the letters.

What is truly amazing about this example is that anyone with a high school education can understand it. If the person has seen any geometry at all, they have seen triangles and right triangles, and if they remember anything at all from their geometry, it is good old Pythagoras.

So now I have a diagram with semantics, and I have moved up a level of abstraction at the "accessibility cost" of moving from the first grade to the high school freshman level of mathematics. Also, at this point, people are definitely intrigued as to where I am going with all this. So I try to bait the hook with a very tasty worm.