Math Mutation 76:  Grue And Bleen

    My wife just looked out the window, and she thinks the grass on
our lawn is green.  But I know better.  I told her that it's really
bleen.  Bleen is a color defined by 20th-century philosopher Nelson
Goodman, defined similarly to the following:  An object is bleen if it
appears green until July 31, 2008, and then blue afterwards.  There is
also a complementary color defined, called "grue", which describes
objects that are blue until July 31, 2008, then green afterwards.
These definitions may sound odd to you, but in my native land of New
Jersey, things changed color from pollution all the time, so these
definitions are perfectly natural.  How can my wife prove that our
lawn is green, and not bleen?  Every day since we moved in, I've
looked out our window, and it's looked bleen to me.
    Being a clever listener, you might object-- "Obviously your bleen
and grue are complicated definitions, not natural notions of color
like the green and blue that we are used to."  But I can easily reply
that you have it backwards.  Earlier I stated the definitions in a
form that seemed complex, but to me, bleen and grue are the natural
notions.  I define your bizarre color "green" as "bleen until July 31,
2008, and grue afterwards."  So you see, it takes lots of words to
construct this artificial color green that you are used to, while my
simple notion of bleen can be stated in one word.  Now, how sure are
you that my lawn is not going to, in your terms, turn blue at the end
of this month?
    This amusing and classic paradox demonstrates the problem of
inductive reasoning in the sciences.  We observe natural phenomena,
try to come up with the simplest explanations and definitions, and
then use this information to model things mathematically.  I'm pretty
sure that at some fundamental level, blue and green are simpler than
grue and bleen-- perhaps the best answer I've heard to this conundrum
is that blue and green can be defined without reference to time, by
describing the wavelengths of light they represent, while grue and
bleen cannot.  But do we always get our inductions right?  Is it
always the case that the simplest explanation of phenomena we observe
will turn out to be the most accurate?
    One well-known example of a case we got wrong is Newton's laws
of physics.  His simple equations describing motion of common objects
and forces seemed right for hundreds of years, until Einstein came
along.  Which do you think would be more suprising:  my lawn turning
blue at the end of this month, or time slowing down a slight amount
for me if I drive my car fast enough?  Yet the latter case turns out
to be true!  What seems like a simple, obvious model of reality is
not always so.  We can never be sure that our inductive reasoning is
right; our careful observations of past events always might be missing
some crucial factor that would lead to a completely different theory.
Take a look out your window-- perhaps your lawn is bleen as well.
    And this has been your math mutation for today.



References:
<a href="http://en.wikipedia.org/wiki/Grue_and_Bleen">Grue and Bleen at Wikipedia</a>