FOM: Some thoughts on "Realism"

[Joe Shipman]

Peter Schuster wrote:

> >To maintain that the twin prime conjecture is indeterminate, you don't
> >have to deny the determinacy of the property  "primeness", nor do you
> >have to deny ontological status to any integers.  You just need to deny
> >that the SET of integers exists as a completed whole.
>
> Doesn't it suffice to reject the idea that all properties/subsets
> of integers are decidable/detachable?
>
> I cannot see why one had to deny the integers "as a completed whole".

Rejecting the idea that all properties/subsets are decidable/detatchable is
not enough, because the particular property being discussed here, primeness,
is a very concrete, computationally straightforward property (strictly
speaking, we are using the property of being a prime which is the successor
of the successor of another prime, but that's just as concrete).

Tennant suggested that rather than rejecting the integers as a completed
whole, one could just deny that the universal quantifier guarantees
determinacy of truth-value.  In other words, we could grant, for each n, the
determinacy of the statement "there is a pair of twin primes above n", but
deny the determinacy of the statement "for all n,  there is a pair of twin
primes above n".  I have trouble with this notion--it is hard to see how
this is not simply denying that the set of integers exists as a completed
whole, because "the set of integers exists as a completed whole" means, to
me, that the set of integers is something you can quantify over with no loss
of meaning.

-- Joe Shipman