Dichotomy

Dichotomy

Either - or

What a wonderfully simplifying and comforting concept it is that things can be bifurcated into black and white, on and off, true or false. Mathematics relies on it. The logic of mathematics requires an object to either be a point or not a point, a line or not a line, a prime number or not a prime number, a glob or not a glob. But even in mathematics, which relies on the Law of the Excluded Middle, things are not always so easy. Without getting into a serious lesson on the foundations of mathematics, there do exist propositions in the context of mathematics which can be shown to have no determinable truth value. In short, math itself as it is currently constructed is not strong enough to ascertain whether or not these propositions are true or false.

How do mathematicians handle this problem? There are a number of ways.

First, we can have everyone pick a side. That's right. Because the proposition cannot be determined within math itself, we are free to accept the statement as true and go on our merry way under that assumption, or we accept it as false and go on our merry way. In doing this, we are accepting the statement (or it's negation) as a new axiom. This has actually happened. The axiom of choice was eventually accepted, not because it's so intuitive (a trait that mathematicians like their axioms to have) but because it was convenient. There was much opposition to the axiom at first, but after a few generations most of it has died because we would lose a whole bunch of cool mathematics without it.

Second, we can ignore it unless we have to pay attention. A methodology not unlike having sex while small domesticated animals are watching. Sure you can take care of business, but every once in a while they might decide you're being playful and want to get in on the fun. This is how mathematicians treat the Continuum Hypothesis. Most mathematicians ignore it as it hardly ever crops up. Others accept it as true, again, out of convenience because it makes infinite sets behave nicely, but others feel there is little reason to accept infinite sets are so well behaved when finite ones don't behave so nicely.

In the end, this option means that number theory isn't really what people think it's cracked up to be. Cracked is an appropriate word because there isn't just one supreme math universe that mathematicians work in, there are many. There is a model to mathematics which accepts the Continuum Hypothesis and another model that does not. Honestly, there are models which do not accept the choice axiom, but they're a barren wasteland in terms of modern research. Physics does the same thing in that it has multiple models that can be used to describe the universe. Most physicists work in what is called the "standard model", just as most mathematicians work within the standard framework of accepted mathematics, but the splinter groups do exist.

Despite this entry being entitled Dichotomy, there is a third option for mathematicians in dealing with a proposition that cannot be determined within the context of mathematics. We can rewrite all of mathematics using different axioms altogether. Axioms that allow us to make a conclusion on these previously undetermined propositions.

Admittedly, this is cumbersome. We've done so well in creating mathematics up until now do we really need to start over? Of course not. One would only need to use their new set of axioms to create the foundations of our previously known knowledge and everything will follow after that. So in the end, it's not really that much work to rewrite all of mathematics.

I guess I'll have to finish this discussion tomorrow.