Formal Indecision


A formal system in mathematics is a system which contains a set of axioms (unquestionable statements of truth), and then adds to these a set of production rules by which new true statements can be generated. The process of production can go on forever allowing a never-ending list of truths to be derived.

A famous result called Gödel’s incompleteness theorem shows that in such a system, there are some true statements that are nevertheless un-provable (undecidable) using the rules of the system. It shows that a self-consistent formal system can never be complete in having access to all of its truths.

There is some sense in which Gödel was lucky in that his incompleteness theorem itself happened not only to be true, but also provable, because it could easily have been a victim of itself. We would then never know for certain if there were true things out there that could not be reached by mathematical reasoning.

This is not unlike Donald Rumsfeld’s famous statement: “As we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don’t know we don’t know.”