FOM: Re: twin primes again
Joe Shipman
shipman at savera.com
Wed Jun 21 12:40:36 EDT 2000
Peter Schuster wrote:
> I understand from your contributions that the
> twin prime conjecture is something different from
> Goldbach's conjecture or Fermat's last theorem.
>
> Do I correctly understand that, according to your opinion,
> no position is possible which simultaneously
>
> (a) does not assume that the truth-value of such
> "highly infinitary" statements as the twin prime
> conjecture is determinated from the outset;
>
> (b) does not deny the whole set of integers as
> a "completed whole", as something "to quantify over";
>
> (c) does not distinguish between statements like
> "for each integer ..." and the corresponding "universally
> quantified" formula?
>
> Note that
> (a) is a crucial point for every constructive philosophy, if
> not for any pragmatic view of mathematics in general;
> (b) is just what I tend to assign to (Bishop's) constructive
> mathematics, although Bishop possibly would not agree;
> (c) seems to be part and parcel of any mathematical practice.
The reason I used the twin prime conjecture rather than Goldbach's
conjecture (Fermat's conjecture has been proven) is that the
independence of the TPC from ZF would say nothing about whether it is
true, while Goldbach's conjecture, if independent, must be true, because
a counterexample could be finitely verified.
This doesn't mean that Goldbach's conjecture is unsuitable as a putative
example of a statement whose truth value is indeterminate, but the TPC
is a better example because of the symmetry between TPC and its negation
(neither would necessarily admit a finite verification). For GC to be
indeterminate would seem to require bringing in the issue of
feasibility, which I'd rather avoid if possible for clarity's sake.
Put another way, I want to set the bar high for indeterminacy, so that I
would like to say, to begin with, "A statement is determinate iff its
negation is indeterminate" and "All theorems of PA are true (and
therefore determinate)". Under these assumptions, "GC is indeterminate"
implies "~GC is indeterminate" which implies "~GC is not a theorem of
PA" which implies GC, an unsatisfactory situation.
I would express your trichotomy positively rather than negatively.
Thus:
One may not simultaneously
a) deny the determinateness of the truth-value of the TPC
b) accept the set of integers as a completed whole which may be
quantified over,
c) interpret the universal quantifier in the ordinary way as "for each
integer n, Phi(n)"
There is an additional linguistic nuance here. "For each" sounds more
constructive than "For all", because one may interpret
(Upside-down-A)n(Phi(n)) as meaning "If we already have n somehow, then
Phi(n)" rather than "For 'all' n, Phi(n)". However, in the presence of
b), "For each" and "For all" seem to mean the same thing.
-- Joe Shipman
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