[FOM] Question on the Scope of Mathematics
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Tue Jul 27 09:50:01 EDT 2004
On Jul 26, 2004, at 9:14 PM, Dmytro Taranovsky wrote:
> However, unless
> stated or clearly understood otherwise, a claim that a mathematical
> statement
> is true includes and should continue to include an implicit claim that
> enough
> is known so that it is "routine" to produce a formal proof of the
> statement in
> the appropriate formal system (typically, ZFC).
But the only reason for assuming that provability in ZFC has anything
to do with
truth of statements is that ZFC is sound. By your account, this either
is not a mathematical
truth, or we should specify a formal system, say MK, in which it can be
proved. But unless
we assume that MK is sound, this again seems to be irrelevant to the
truth of the claim
that ZFC is sound. We would have to specify a formal system in which
MK's soundness
is provable, but establishing this would require yet another formal
system and so forth.
Thus it seems that by your account stating the truth of a mathematical
statement is not itself
a mathematical statement, which seems a bit odd.
--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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