Re: Epistemology 201: The Science of Science
From: Jason (jasonstevensNOSPAM_at_free.net.nz)
Date: 01/25/05
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Date: Wed, 26 Jan 2005 11:41:40 +1300
> > If maths develops according to the way we see the world working, then the
notion
> > of mathematical 'truth' can be extended from the limited formal system
notion of
> > 'truth', to our every day understanding. I believe there are mathematical
> > statements that are true in our world, but are not provable by the current
> > axioms of maths. These statements are the ones I am referring to. They
need to
> > be empirically verified, which might then provide impetus for the axioms to
> > change.
>
> Could you provide an example of such statements? The only truths not
> provable in the current axioms of math that come to my mind are
> artificial examples like Goedel sentences, which we believe to be true
> for theoretical reasons (they'd better be true, or arithmetic is
> inconsistent), not on the basis of anything in our experience.
Unprovable hypotheses about Prime numbers spring to mind. The Riemann
Hypothesis has a lot of empirical weight, but has never been proven. I don't
think this one has been shown to be outside of proof though. The Continuum
Hypothesis has been shown to lie outside of mathematical proof. But
mathematicians don't wait around for proofs, they use these hypotheses and
qualify that their results are dependent. They use them on empirical grounds
because by all counts they look correct.
> > Exhaustion and testing by algorithm is not so formal. The computer has to
be
> > trusted to be doing the right task. I don't know about yours, but my
computer
> > is not to be trusted at the best of times. Let alone the software I
write...
>
> So a method is only formal if it is applied by an infallible reasoner?
> Well, that's an easy way to rule out anything from being formally
> proven, but it's quite a non-standard use of the term.
It is an ideal. And yes, a non-standard use of the term. But then this is a
philosophy channel so we're allowed to explore the limits of concepts.
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