[FoRK] Re: Anything to be learned from religion?
Stephen D. Williams
Fri Aug 12 03:52:02 PDT 2005
Wilkin, Kurt wrote:
>Axioms are unprovable (they were originally 'self-evident truths').
>Standard example : how many parallel lines are there through a point?
>Different values of which will give you (roughly)
>1) the sum of the internal angles of a triangle always = 180 degrees
>b) the sum of the internal angles of a triangle sometimes != 180 degress
>(the example I got for b was that if you *could* sum
>the angles between 3 atoms, and between 3 universes,
>you'd probably find a difference. Think a better one
>is a triangle on a sphere)
Those don't prove the point at all. You may be pointing out the
incompleteness of the specification of the axiom, but it's still a
theory that either can be proven wrong or it can't. Conjecture about
whether it might be proven wrong if you could do some out of reach test
at the quantum level is no different from any other scientific theory.
(Not to mention that it is current theory that things are just different
at the quantum level.)
The example of the triangle on the surface of a sphere is similarly the
application of an axiom with implied qualifications to a situation where
one of those qualifications is broken. That doesn't change the essence
of what the original theory was or whether it was true or not, it is
simply interesting due to the pendanticism of mathematicians.
Saying axioms are unprovable really means two things I think: A) except
in closed systems of logic, no statement can really be proven only
disproven, B) there might not be a complete enough closed system of
logic to build a proof. In science, you almost never have a closed
system of logic or anything similar. You make a model that fits the
facts and throw it away when you are able to replace it with a better
model (or you can't fit any model).
I think that, just like the meaning(s) of "theory", "prove" and "truth"
have different shades of meaning.
Truth is really the best, simplest (a la Occam's Razor), most
consistent, and least disproven theory that has some solid evidence.
If I create an axiom that says that adding a positive integer to an
integer makes it bigger than it was before, it is pretty easy to decide
that this is "truth".
Conversely, if I give you a lava-lamp random generator and tell you that
it is providing direct yes/no decisions from God, you can determine that
this is not likely to be true even though you can't disprove a carefully
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