FOM: Simpson query

[hartry field]

On March 3 Steve Simpson wrote:
> Maybe Field is really agreeing with me and I'm just not
>understanding some other subtle point that he is making.

I don't know about 'subtle', but yes, I am agreeing with Steve and making
another point.  (I acknowledge responsibility for the confusion: in my Feb
27 posting I didn't say quite what I meant, though I thought that the Mar 2
posting clarified it.  Evidently not.)  The point I was making rests on a
distinction between truth and determinate truth.  To be a bit more explicit
about the background for this, it seems to me that in reasoning in languages
with vague or otherwise indeterminate predicates, we ought to apply
classical logic and classical truth theory; so e.g. either 'George is bald'
or 'George is not bald' will be true, even when George is a borderline case.
But if George is borderline it won't be determinate which is true--neither
will be determinately true.  I think that this is relevant to mathematics,
because I think that notions like 'set' have a kind of indeterminacy that is
in important ways like vagueness: nothing in our practice with 'set' or
'member of' could have ruled out all the nonstandard models of set theory as
"unintended", so many formally undecidable sentences (eg about the value of
the continuum) will be not only formally undecidable but objectively
indeterminate.  They are still either true or false, by classical logic plus
classical truth theory, but not determinately true or determinately false.

In the case of number theory I am reluctant to believe that sentences
formally undecidable in the relevant theory (call it M) are objectively
indeterminate.  [Here I suppress a complication about there being no one
unique relevant theory, which leads to an analog of higher order vagueness.]
In the paper that Torkel Franzen cited that led off this discussion, I
offered an explanation of how our practices with 'finite' might make it and
'natural number' determinate despite the nonstandard models.  But the
explanation relies on empirical assumptions that could be wrong, so it is
important to accommodate the possibility (call it View W, for 'weird') that
ALL formally undecidable sentences are objectively indeterminate, even the
PI-0-1 ones.  If so, then again by classical logic and classical truth
theory, such sentences will be either true or false, but not determinately so.

I took Charles Silver to be implicitly offering an argument against the
coherence of this: take a PI-0-1 statement G; if it is undecidable in M,
it's instances must be true, so G must be determinately true; contrary to
the hypothesis that nothing formally undecidable in M can be determinately
true.  (Silver said undecidable in PA, not in M, but really it's
undecidability in M that's relevant.  He also said 'true', not
'determinately true', but again it's determinate truth that is relevant to
View W.  Not making this clear was one of the problems with my Feb 27
posting.)  Goldbach's Conjecture was the G he picked.

Part of Steve's point is that if GC is provable in M to be PA-undecidable,
that's an M-proof of GC, so GC comes out determinately true (even on View
W).  I agree (and said so in the posting of Mar 2 to which Steve is
responding).  That case isn't relevant to my point, for again, it's
M-undecidability that would be relevant.  (And of course it won't be
provable in M that GC is M-undecidable, unless M is inconsistent.  For
present purposes let's suppose that GC isn't determinately true when M is
inconsistent.)  

The more relevant case is where GC is M-undecidable (and hence
PA-undecidable), but not M-provably PA-undecidable.  My claim in this case
is that (assuming View W), GC is not determinately true.  Now, I grant that
if GC were DETERMINATELY PA-undecidable it would be determinately true, as
Steve says. However, (on View W) its DETERMINATE PA-undecidability requires
its M-provable undecidability, which is contrary to the present case.  So I
see no problem with View W here.

You may well ask: what can it mean to suppose that GC is M-undecidable but
not determinately so?  I grant that we wouldn't want to assert that GC is
M-undecidable if it weren't provable (which is equivalent to, 'without its
being determinately true', on present suppositions).  But we can assert 'GC
is M-undecidable or it isn't' (since we're keeping classical logic in the
face of indeterminacy); and it is important that the first disjunct leads
(on View W) to the conclusion that GC is not determinately true.  

The view being suggested here is NOT, I repeat, one I find at all
attractive; but I do think it is coherent.  (One unattractive feature of it
is that it entails that M can't be DETERMINATELY consistent.  I discuss this
in the paper Franzen cited.) And not only coherent, but interesting, for I
think that unless one can explain what it is about our use of 'finite' and
'natural number' that makes them determinate, there is a strong case that we
are forced into it. 

I hope that clarifies things.  Sorry for being longwinded.-- Hartry Field