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.
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.
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