The Existential Mathematician.

As contradictory as it seems one of the most unique proofs of the 20th century, provided by Gödel, formally shows that no axiomatic system (an example being mathematics) can be “complete”. With respect to math this means no theory built on axioms can prove all theorems, or worded differently: nothing built on finite rules can prove all the questions which it can create. Gödel’s proof is done by constructing an abstract language that all axiomatic languages must inherit from, then it is shown that this ‘language’ must have statements that are impossible to prove using that system. More specifically he shows that sentences built from finite elements (where in math sentences would be theories) can be proven or disproved. To better explain here is an example:

“Everything I say to you is a lie.” This sentence is undecided. I am neither a liar nor do I speak the truth. If you take my sentence as true, meaning you believe “all I say is a lie”, then the sentence is false, but it is false because you took it as true. If you take the sentence as false, as a lie, then what I say is the truth. Like mathematics your decision skills are incomplete.

In short:

Gödel has shown that mathematicians are painters with more paper than ink. There can never be a masterpiece.

-cityofsalt