With his two Incompleteness Theorems, the late mathematician Kurt Gödel "showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved." So Carl Boyer explains in his History of Mathematics.
Even as non-mathematicians may not understand Godel's logical formalisms, we may well sense that the theorems function poetically as well as mathematically. They are an invitation for all of us to accept the incompleteness of life: the fact that, no matter how certain we may feel about our convictions, there's always something more, something novel, something incomplete. This is not bad news; it is good news. It means that life, like pure mathematics, is an adventure.
- Jay McDaniel, 11/20/2