FOM: Some thought on "Realism"

[Joe Shipman]

Tennant wrote: "The recent discussion on fom about ontological realism
and realism about truth-value is whether the former *implies* the
latter."

I am having trouble understanding the discussion.  It would really help
to fix ideas.

First of all, is there any distinction between "ontological realism" and
"Platonism"?  I have always understood "realism", as a philosophy of
mathematics distinguished from Platonism, to be "truth-value realism" --
the position that mathematical sentences have a truth-value which we may
or may not discover but which is independent of us.  (There is actually
more than one flavor of "truth-value" realism, because the sentences may
be <1>in the language of arithmetic or <2>in type theory or <3>in the
language of set theory; the third of these implies the first two, the
second is all that is needed in normal mathematics, and the first is the
most fundamental.)

Naively, if one interprets the backwards-E quantifier as "there
exists",  truth-value realism implies ontological realism.  Can someone
suggest an alternative ordinary-language interpretation of this symbol
that does not trivialize this implication?

Tennant asked about the other direction.  To fix ideas, let's take the
simplest statement that could conceivably be indeterminate in
truth-value, the twin-prime conjecture, "For all x, there exists y such
that x<y and y is prime and (y+2) is prime.", which is pi^0_2.  (Does
anyone out there want to suggest a pi^0_1 candidate?)

I hope no one will deny that the property of being a prime is
determinate -- even if some integers like Friedman's n(3) can be denied
ontological status because they are too large for the Universe in some
sense, if we HAVE an integer n somehow it must be determinate whether it
is prime.

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.

I can imagine that the twin prime conjecture is radically, absolutely
unknowable if I deny that there are any actually infinite objects or
minds.  At that point it becomes a matter of taste whether you say it
has no truth value or whether it has a truth value that no one can ever
know.

So ontological realism implies realism about truth-value in the sense
that sentences which quantify over members of a set have a truth value
if the set (not just the individual members) exists ontologically.  Put
another way, "full" ontological realism (whatever that is) implies
truth-value realism, but ontological realism "up to kappa" doesn't imply
truth-value realism for statements quantifying over kappa.

It also seems clear that truth-value realism is a coarser notion --
there are at most continuum-many possible truth-value realities, but
many more possible ontological realities (if there are more than
continuum-many inaccessibles, there will be a pair with the same true
sentences for their associated "universes").

Consider the set of complete theories in the language of set theory that
include ZFC.  This is a well-defined set of real numbers, by a
straightforward coding.  "Full" truth-value realism is the position that
this set has a distinguished element.

One last thought:  "The number of even digits in Friedman's number n(3)
is greater than the number of odd digits" seems, in one sense, even more
strongly unknowable than the twin prime conjecture; the preceding
remarks have been ignoring feasibility, so that "unknowable" in this
practical sense should not be confused with "indeterminate".  I don't
want to dismiss the philosophical importance of feasibility, I just want
us to clarify as much as possible issues related to infinity before
bringing it in.

-- Joe Shipman