-- E. T. Bell
In the last chapter we were introduced to the concept of betweenness of points, which is maybe the first time so far that we've smacked our collective foreheads and said, "No kidding. Do we really have to waste our time defining this idea that's so totally obvious?" On some level, this is a legitimate complaint. I mean, we can look at a line segment with P and R as its endpoints and immediately understand that every other point on the segment is between those two. This picture completely jibes with our everyday understanding of what it means for a thing to be between two other things. And in that world, betweenness doesn't seem confusing in the least. You know that your C- in Geometry is somewhere between an F and an A (or between a D and an B, for that matter). You know that my classroom is between Mrs. Brown's and Miss Forsberg's (and so is Mrs. Wavrunek's). And you know that the Vikings are somewhere between awful and terrible. These are facts.
Once we start talking about geometric ideas, though, we need to be very, very careful about things that seem to be obvious at first. I'll tell you that this is a difficult thing to do. In fact, Euclid, who is generally considered to be the LeBron James of Geometry, was pretty terrible about assuming 'obvious' things that turned out to be not so obvious--and were in fact sometimes just plain wrong--and it got him (and everybody else) into all kinds of trouble down the line. It took a few centuries' worth of effort by lots of mathematicians to tidy up all of the things Euclid and others took for granted as 'obvious.' This betweenness business is one of the untidy loose ends in question.
The truth is, we're going to run into a lot of this sort of thing this year. Whenever we feel that something is intuitively obvious and true, our first instinct should be to define it precisely so that we don't let anything slip through the cracks. It can be a little tedious, but we need to make sure that we're on sound mathematical footing so that we know our next steps won't leave us hanging in the middle of the air, logically speaking.
"Wait, wait, wait," you say. "We've already made some totally unfounded, obvious statements like, 'Any two points define a line.'" Okay, you got me. We're not going to be able to get away without any appeals to our pure intuition. The way we've chosen to teach you Geometry (along with something like 99% of high school students in the U.S.) is axiomatically. What this means is that we start off with some basic definitions and principles, offered completely without proof or justification, and then build off of those, very carefully and logically, until we come up with all kinds of interesting and useful results. These basic principles are called axioms, and they simply define the rules of the game. (So we need a few unfounded statements, but those should be the absolute bare minimum assumptions necessary to define our Geometry for us. In fact, the Euclidean Geometry we'll spend all year studying has only five.) We need to avoid thinking of the axioms as "true" in any normal sense of the word; they're just things that we assume in order to have some basic building blocks for our Geometry.
Here's an example. So far we've assumed that everything we've studied has taken place in the plane (truth be told, a very particular sort of plane). We look at pretty diagrams drawn in that plane and lots of things seem obvious to us because we think that the plane somehow represents a kind of reality. If we really think about it, though, we probably shouldn't be starting with the plane at all. The surface of the Earth that we crawl and walk and run around on is actually something very like a sphere (but not quite), and spherical geometries have quite different properties than planar ones. To take just one strange example, the interior angles of triangles almost never add up to 180 degrees on a sphere. In fact, we can draw triangles that have three right angles (can you think of how?)! Any piece of the surface of the Earth, no matter how small, is actually curved, so why do we think that shapes drawn on flat sheets of paper are somehow more realistic? More true? We shouldn't, and they're not.
Anyway, here's the point: obvious is indeed a dangerous word, because it implies that an idea doesn't require any scrutiny, that it's somehow above suspicion, and as mathematicians we are in the business of being suspicious. We have to scrutinize every conjecture and hypothesis, because that's the only way to separate truth from fiction. We're really just scientists with cooler calculators. Obviously.
