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Re: WAS-- Re: Hanson 2006, Mortimer, Baeker response

On Sun, Jun 25, 2006 at 09:04:49AM +0000, Phil Bigelow scripsit:
> "Andreas Johansson" <andreasj@gmail.com> writes:
> > >  The axioms, themselves, are based on earlier real-world
> > >  observations.
> > Not necessarily.
> I noticed that you didn't say "never".  You have this wonderful
> ability to under-elaborate on your responses.  ;-)  Give us an example
> of an axiom that is based on earlier real-world observations.

"The sum of the interior angles of a triangle is 180 degrees."

Since you may well have meant *not* based on early real-world
observations, consider "the sum of the interior angles of a triangle is
less than 180 degrees"; that'd apply to concave-plane geometry, of which
we don't have a real-world example.  ("Greater than" gets you into
convex plane and spherical trig used for navigation.)

Advanced math -- eg., Riemann spaces with all the bizarre, n-dimensional
stuff involved in manifolds and tensor transforms -- frequently hasn't
got any specific connection to *anything* tangible.

Practical applications of math, including statistics, represent either
isomorphic (point-to-point) or homomorphic (math sense; "A
transformation of one set into another that preserves in the second set
the operations between the members of the first set."  So "descended
from" in the model is the same as "descended from" in evolution, sort of
thing) models, and *of course* those relate to real-world things in some
relatively clear way.

Most of math isn't like that.

I mean, come on, if you're talking about measuring properties of
arbitrary curvature in arbitrary dimension, you don't have anything of
arbitrary dimension to point to, do you?

The thing about the models is interesting for dinosaur stuff because
there's absolutely no way we can have an isomorphic model -- no soft
tissue characters, no genes -- and we *probably* can't have a
homomorphic model based solely on osteological characters, but proving
that one way or the other would be astonishingly challenging.

So one of the potential challenges for phylogenetic systematics would be
to find a way to quantify how bad the model is.

-- Graydon