Parallels

Historically geometry has been the chosen vehicle for teaching sound logical reasoning. You could start with a few undefined entities and postulates and derive a very useful system of theorems, which in turn could be used as shorthand to develop other theorems, and so on. The results aren't merely useful, but can be very intellectually satisfying.

Sometimes they're too satisfying. There was no obvious endpoint to what could be known by using logic to extend existing knowledge. And if there was no such limit, logic tells us that eventually, with enough work, we could know as much as there was to be known about geometry. Maybe it's no coincidence that the culture that gave us so much of our geometry, the Greeks, also gave us the concept of hubris.

Geometry was born as a means of solving practical problems. It was found that rules derived logically from geometry also could be used to solve practical problems. This invited the conclusion that the real world behaved by logical rules. That in turn implies that if we work long enough, we'll find them.

There are problems with those idea though. It so happens that we can construct consistent logical systems in many ways. So even if we assume that the universe behaves by logical rules, we still don't know what set of rules to use. For an example, consider non-Euclidean geometry.

When Euclid constructed his geometry, he sought to do so with a minimum number of undefined entities (points, lines, planes...) and postulates. Then he got to what is called the parallel postulate.

In essence, Euclid's parallel postulate says that given a line and a point not on the line, exactly one line exists which contains the point and is parallel to the given line. This seems intuitively logical and appears "right" in plane geometry.

Euclid didn't like it though. He thought he ought to be able to derive it from other postulates and theorems. He never succeeded in doing so, and thus concluded that because it appeared to be true, he had to include it as a postulate if he wanted to use it.

Pragmatically this was a sound decision, but was it sound logically? Yes and no. Yes, because it leads to useful results. No, because it is entirely possible to construct a consistent, useful geometry without doing so. The resulting geometries are called non-Euclidean geometries.

One non-Euclidean geometry replaces the parallel postulate with one that says there are many parallels. Another says there are none. Both "work", and have practical applications.

OK, which one is "right"? Wrong question. You use the right one for the job. For most of us that's almost always Euclid's geometry.

What's important here is to realize that abstract logical systems like geometry are one thing, and real life is another. Science is about bridging the two - building logical systems that behave like real life.

But we can't ever know what is "right", only what is "better".