Anything to be learned from maths,
was RE: [FoRK] Re: Anything to be learned from religion?
Wilkin, Kurt
Kurt.Wilkin
Fri Aug 12 07:26:56 PDT 2005
on Friday, 12 August 2005 10:52 p.m. Stephen D. Williams wrote:
> Wilkin, Kurt wrote:
>
>> ...
>> Axioms are unprovable (they were originally 'self-evident truths').
>> Standard example : how many parallel lines are there through a point?
>>
[...]
> Those don't prove the point at all. You may be pointing out the
> incompleteness of the specification of the axiom,
Rather, the point of axioms being unprovable is that at some
level, your proof will depend on some other axiom which must
then be part of the system within which you are proving
that axiom.
And if you can prove that axiom, you'll depend on some other
axiom which, again, is part of the definition of the system
within which you're proving or disproving the statement.
And if you've proven an axiom, you've at least proved that it
isn't an axiom.
> 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".
If I also create an axiom that says that adding a positive
integer to an integer doesn't always make it bigger, it is
pretty easy to decide that this is also "truth",
but that you are talking about arithmetic, and I'm talking
about modulus or clock arithmetic : its the axioms that
define what we are talking about.
That mine is necessarily false in your system doesn't make
your system any less complete (or incompletely specified)
or make mine false, and vice versa.
Cheers, Kurt.
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