Brent Meeker wrote:
> Tom Caylor wrote:
> > 1Z wrote:
> >> Tom Caylor wrote:
> >>>> Again, the kind of formalism that says
> >>>> everything can be brought under a single
> >>>> formal scheme (the Hilbertian
> >>>> programme) is different from the kind
> >>>> that says mathematical truths are dependent on axioms,
> >>>> and different truths will be arrived at under different
> >>>> axioms. Of course the key point here
> >>>> is "different truths". Tom is not entitled to assume that
> >>>> all roads lead to Rome.
> >>> If your definition of truth is limited to logical inference given a
> >>> certain set of axioms and inference rules, then what are we trying to
> >>> do on the Everything List?
> >> That's *mathematical* truth.
> > Mathematical logic is richer than that. This is what Bruno is saying,
> > that the math path points toward Rome. And it is no more scary (a la
> > possible spirits lurking under/in every rock) than the matter path.
> > Limiting math as you are doing, and as Brent Meeker does in his
> > response to my (X and not-X) note, is ignoring such evidence as the
> > proofs of Godel and Tarski's Indefinability Theorem.
> > Tom
> But as I understand it, Bruno wants to identify "mathematical existence" with
> true existence statements about mathematical objects.
> Some of these are unprovable. They can be consistently added to the axioms.
> But also their negation can be consistently added to the axioms. But not
> both. So there are disjoint realms of consistent mathematics - some are
> Rome, some are Athens, some are Tuva, most are Harry Potter's home town.
I suppose that is why he wants to base his case on the
subset of arithmetic everyone agrees on.
> Brent Meeker
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