A Gödelian meta-sign, or, Do you follow your own advice?

On my daily walk yesterday I saw this sign:

This was on a weird little plot of ground at a high-traffic intersection. And indeed, unlike everywhere else along the road in Portland, this patch featured no signs for the upcoming election, gutter cleaning, martial arts classes or anything else. So I guess it was working.

A few meta-thoughts

Would the sign have been equally effective if it had included the words “Except this one” ? Or is it actually more effective since it draws attention to its own self-contradictory nature?

In graduate school I was studying to be a mathematical logician. Logicians don’t actually study how to reason (that’s in the introductory course and it is considered trivial — they leave it to the philosophers). No, mathematical logic is the study of systems of logic. One of the giants of the field, Kurt Gödel, managed to prove that if a system of logic is consistent (that is, doesn’t contradict itself) and is powerful enough to include basic arithmetic, it can encode a statement that basically says, “This statement is false.” You cannot prove such a statement; if you could, you could prove a false statement, which violate the consistency requirement. You can’t disprove it either, since then it would be true, and thus would be false, an inconsistency. Gödel’s formalization of this argument showed that in any reasonable mathematical system, there are always statements you can neither prove nor disprove. This yard sign is a pretty good reminder that you can do the same thing in the real world, too.

Finally, you have to laugh at this sign, since it so simply encapsulates the hypocrisy of the person who put it there. But take a moment to think about yourself. Are you making complicated arguments about how the best arguments are simple? Are you insisting on your freedom to say that some kinds of speech much be prohibited? Are you bullying people with your anti-bullying messages?

Consider your most firmly held and widely shared belief. Now think about how often and in what ways you violate it. Because we all do. You’re better off when you start by admitting the flaws in your own consistency. It’s the Gödelian thing to do.

We’re all hypocrites. Except me, of course. I’m 100% perfectly consistent.