From: Paul Fidika (Fidika@new.rr.com)
Date: Sat Aug 14 2004 - 01:56:34 MDT
John K Clark wrote:
> >after Gödel had just proved that the Continuum Hypothesis
> >could not be disproven within Zermelo-Fraenkel Set-Theory,
> >he went on to give some arguments for why he believed that
> >the Continuum Hypothesis was FALSE! Now, some people might
> >say that Gödel should not be allowed to get away with this;
> >they believe that the truth or falsity of a statement
> >should only be considered relative to some formal
> >system of axioms,
> The Continuum Hypothesis is either true or false. Of course it could
> also be un-provable, if it is that means you will never find a
> counterexample to show it's wrong and you will never be able to prove it
> in a finite number of steps to show it's correct. However I don't know
> anybody this side of a Looney bin who claims the truth of falsehood of
> something depends of the formal system of axioms used.
> Will this bridge collapse if I drive my heavy truck across it? Well,
> under some formal systems it will and you will die and in others it will
> not and you will live.
> I don't think so.
Actually I was only referring to the truth of falsity of a statement in a
formal sense, and the viewpoint I was describing (which is not so common) is
called Intuitionism (e.g., see http://en.wikipedia.org/wiki/Intuitionism).
However, I suppose the position could be defined analogously in a more
informal setting as well--we could say that a statement is true if we could
(at least in theory) construct some experiment which would confirm its truth
or the truth of its negation. Now consider the existence of parallel
universes--if there is no experiment which can prove or disprove their
existence, then these generalized-intuitionists would say that the statement
"parallel universes exist" is neither a true nor false statement--it's
nonsense. The Platonist would say that either parallel universes exist or
they do not, although we may never be able to determine which is the case.
Despite Intuitionism's name, I find Platonism the more intuitive of the two.
This topic seems to comes down to: did humans invent Mathematics or discover
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