[FOM] Platonism, "vagueness" of N, possible inconsistency of PA?

[Randall Holmes]

Dear FOMers,

     Vladimir Sazonov asks how platonism could survive a proof of
~Con(PA)...

     Platonism doesn't necessarily require a commitment to any
particular mathematical content; what it involves is a particular kind
of attitude to mathematics.  A platonist believes that mathematical
statements have well-determined truth values (independent of our state
of knowledge) and perhaps that there are special mathematical objects
to which they refer (though I use second-order logic to apparently
avoid the need for special mathematical objects, some may insist on
regarding such a use of second order logic as involving special
mathematical objects anyway -- second order variables can be
understood as referring to sets).

     If I were in fact able to prove ~Con(PA) (which I do not seem to
be able to do at the moment, much to my relief :-) I would not
repudiate the idea that that mathematical assertions have specific
truth values; I would conclude that certain statements which I had
believed to be true were incorrect.  In particular, certain statements
normally construed as being about the "standard model of the natural
numbers" (though they can be interpreted without the need for any
canonical such model) would be revealed to be content-free because
there would be no such model or models.

     I don't see any vagueness at all in the reference of statements
about N.  If there is a relation which associates each object in its
domain with exactly one object in its range, which has every object in
its range also in its domain but which does not have every object in
its domain in its range, and if it is possible to define properties
impredicatively (define properties using quantifiers over all
properties) then every statement of the usual language of second-order
arithmetic can be assigned a perfectly clear meaning (because
impredicative second-order logic would then allow one to define a
model of this theory, though not a canonical one).  A proof of
~Con(PA) would show (among other things) that impredicative
second-order logic on infinite domains is inconsistent, and so that
the conditions I state above do not hold; from my standpoint, it would
show that N does not exist (and also that the collection of all items
of syntax (as usually understood) does not exist).  I would not arrive
that the conclusion that the usual idea of N was "vague", but that it
was wrong (clearly defined but resting on incorrect assumptions).

     I don't (on reflection) find this outcome _inconceivable_.  I
have returned to thinking it unlikely, and the consequences would be
unpleasant (which is not in itself a reason not to investigate the
possibility).

     In this situation, we would need to reformulate mathematics using
some other notion (probably multiple  notions) of "natural number" and
"item of syntax".  There are predicative formulations of these ideas,
for example, along the lines of Russell's ramified theory of types.
Such systems are certainly worth investigating for their own sake, and
so are systems for implementing strictly "feasible" mathematics.

     Depending on the status of impredicative second-order logic, the
relation between mathematics and metamathematics would be different.
In second-order logic, one can successfully refer to the syntax one is
using (and so one cannot have complete proof techniques).  If
second-order logic on infinite domains were inconsistent (a
consequence of ~Con(PA)), then one could not do this: it appears
likely that if one were doing metamathematics the syntactical objects
to which one would be _referring_ would actually have to be different
from the syntactical objects one was _using_, which would require
extraordinary care.  (Of course, such care is always required if one
is restricting oneself to the expressive capabilities of first-order
logic, and always advisable for the sake of clarity in any case) [I
don't want to reopen the "is second-order logic really logic?" thread;
in this paragraph I stipulate that second-order quantifiers are to be
understood as ranging over _all_ properties or relations of suitable
arity (which may be understood as sets if one prefers) on the domain
of objects referred to by first-order variables.]

--Randall Holmes