re: #755 Salamantis

Principia Mathematica has been falsified by Kurt Godel, who proved that any logical system complex enpugh to permit self-referential statements must contain undecideable propositions, and therefore must be either incorrect (contain untruths) or incomplete (exclude truths).

Godel’s Incompleteness Theorem is breathtakingly simple. First we postulate Axiomatic system A, and state that every true statement, and only true statements, are contained within it. Then we create Statement B, a statement that possesses an interesting quality - self-reference; it talks about itself. And what it says is “B is not an axiom of A.” What has happened here?

If we include Statement B within Axiomatic System A, then A now contains the false statement that B is not an axiom of A, but if we exclude Statement B from Axiomatic System A, then the true statement that B is not an axiom of A lies outside of A, and A therefore does not contain all true statements. B either belongs NEITHER inside NOR outside of A, or it belongs BOTH inside AND outside A (both impossible alternatives, involving the (neither X nor not-X) and the (both X and not-X) logical contradictions, respectively), and the contradiction is not resolveable within Axiomatic System A. The bottom falls out; mathematics is revealed to be fundamentally a Zen koan.

The bottom does not fall out. Mathematics is the art and science of knowing something about what logically follows from what. What Goedel observed is that mathematics is not powerful enough to let us know EVERYTHING that is mathematically true. There will be some things that are true even though we cannot prove them.

That’s par for the course. There are physics observations we cannot make. Chemistry experiments we cannot conduct. And the reach of human folly and majesty, we aren’t going to understand either, because that too is self referential. If we were bright enough to understand each other, we’d be too bright to be understood by each other.