Can you judge your own sanity?

Uncovering Peculiar Logic

“Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ... How can you figure out if you are sane? ... Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar' or not, given that you have only your own logic to judge itself? I don't see any answer.”

-Hofstadter, Gödel, Escher, Bach

The Incompleteness Theorem

Kurt Gödel’s incompleteness theorem may strike you as unsettling. It states that math, or any other logical system based on a finite set of axioms, cannot answer its own questions. It contains statements that are neither true nor false and true statements that cannot be proven. Gödel’s basic argument can be summed up by the following two self-referential mathematical statements.

The first one is “This statement is false”. It is called the liar’s paradox, and it is true if and only if it is false. Therefore, it is neither true nor false. So math contains statements that cannot be classified as true or false.

The second one is “This statement is unprovable”. If it is true and provable, then you are proving a falsehood. This is generally considered impossible, so the statement must be true and unprovable.

The Limits of Math

So math contains things that are true but cannot be proven. This rivaled David Hilbert’s notion that all of mathematics can be deduced from some finite set of principles. Math, and for that matter any logical system, has and will always have its limitations, in the sense that there will always be more true statements than can possibly be proven.

Further Implications

"Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ... And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself”

-Jones and Wilson, An Incomplete Education

A Little About Gödel

Kurt Gödel, born in 1906, was a smart inquisitive child with hypochondriac preoccupations. In 1927 he met a dancer in a Viennese night club named Adele, and they began a relationship despite parental disapproval; Adele was 6 years older than Gödel and had been married before.

Kurt Gödel was only 25 in 1931 when he developed and published his incompleteness theorem and greatly impacted 20th century mathematics. In 1938 Gödel and Adele were married, and the next year they fled Nazi Germany. They settled in Princeton, New Jersey, where Gödel worked with Albert Einstein.

Later in his life Gödel became increasingly paranoid about the spread of germs. He was known to compulsively wash eating utensils and wear ski masks with eye holes everywhere he went. He also became convinced that his food was being poisoned, so his wife Adele became his food taster. Kurt Gödel died in hospital at age 72 from a combination of digestive complications and his refusal to eat.