by Winkie on Wed Dec 22, 2004 5:59 pm
I'm not satisfied with his explanation of Godel. He's taking it to be all about relativism. I'm not saying that relativism has no place in mathematics -- it's kind of the basis of non-euclidean geometry -- but it's not what Godel's Incompleteness Theorem is about.
The essence of Godel's Incompleteness Theorem is this: No system of mathematical proof is complete. For every system of proof, there are things that are undecidable - things that the system in question can neither prove nor disprove. Other systems might have those undecidable propositions covered, but they'll have gaps of their own. It's not that there's no absolute truth, it's just that no system can describe the truth completely.
And this isn't just hippie philosophizing, either. Godel provides a recipe for constructing a statement that's true but not provable in a system of proof, given the system's rules. What it all comes down to is using the system to model itself mathematically, so you can construct statements about what the system can and cannot prove, and put those statement within the scope of the system itself. This way, you can construct a statement G that essentially says 'Statement G is not provable in this system', and there's your undecidable proposition. Note that G's undecidability does't mean that it's neither true nor false - in fact, because what it says is that it can't be proved within the system, its undecidability makes it true. But we only know that because we're reasoning outside the system.
There are life lessons we can draw from this. Mainly, humility. Godel reminds us that no one knows everything, not even in a field as sharply-defined as number theory.