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I’m dreaming of a white… lie

December 25, 2014

All across America, millions of parents conspire to tell their gullible children the same lie, tricking them with fake incentives to behave in a way that conforms more to authority.  Sounds horrible right?  Actually, it is generally considered harmless!  (And also, it’s not religion!)  A big, fat lie.  Do you know what it is?  If you still haven’t figured it out, does the title of the post jingle any bells?

We tell children that Santa makes a list of “naughty” and “nice” children, and gives only the nice ones presents.  There is no naughtiness continuum – it’s black and white, much like hell and heaven.  In theory, if children believe that they could be either naughty or nice, they will act nicer, hoping for presents.  As a matter of fact, the children are probably getting presents from “Santa” regardless.  And of course, some of these kids are actually relatively naughty.  But they probably believe they are “nice”, at least on Christmas.  And indeed, the promise of presents probably makes them behave a bit nicer, so the parents (err, I mean Santa) win.

Presumably the lie is mostly harmless, makes children and adults alike happy, and, most of all, is part of our culture.  Also, many kids probably never believed it, and certainly not for very long.  That said, in light of the holiday spirit, I’m going to celebrate a virtue of having uncertainty about things that are false (hey!  I *am* nice).  Let me explain, with a lie:

  1. Let L be the sentence “If L is true, then Santa exists”
  2. Suppose L is true.  Then, “L is true” is true.  So since L is true, by its definition, we have that “Santa exists” is true.  So we have proven that “If L is true, then Santa exists”.   So we have proven L!
  3. But again, L being true implies Santa exists.  So we have proven Santa exists!

Take a minute to digest that argument.  Notice this argument works for “Satan exists”, or “pigs fly”.  If you’re curious, this argument is closely related to Löb’s theorem, which I learned about in college from LessWrong and some friends.  The general takeaway is supposed to be “Just because something is provable doesn’t mean it’s true”.  Does that mean we shouldn’t believe things are true when we prove them?  No, it just means that just because you believe something doesn’t mean it’s true.

Another closely related theorem is Gödel’s second incompleteness theorem.  It states that if a formal theory proves its own consistency, then it is inconsistent.  Thus a consistent theory should remain agnostic about its own consistency (It also can’t assert its own inconsistency, of course, since it would be wrong about that.  Or would it?)

As a human, I believe I’m relatively inconsistent.  But that doesn’t mean I spout off nonsense at every opportunity (I mean, maybe I do, but that’s not the point).  I still aim to be consistent, and as a result, am somewhat more consistent.  The Santa Claus lie greatly resembles this, if you replace “Truth” and “Falsity” with “Niceness” and “Naughtiness”.  Naughty or nice, a kid should believe they can be nice, and by doing so, they become a bit more nice.  In reality, they’ll never be perfect in the eyes of their parents (assuming the parents are reasonable), but at least they can get some presents.

From → Math, Random

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