FOM: Some thought on "Realism"
neilt at mercutio.cohums.ohio-state.edu
Thu Jun 15 18:18:32 EDT 2000
Joe Shipman wrote:
> First of all, is there any distinction between "ontological realism" and
I think Platonists sometimes run together the two strands of ontological
realism and truth-value determinacy, without any concern to distinguish
them. But if I had to choose, by way of new convention, what "Platonism"
should henceforth mean, I'd reserve it for ontological realism.
> 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.
So have I.
> Naively, if one interprets the backwards-E quantifier as "there
> exists", truth-value realism implies ontological realism.
Arguably, not so on a substitutional of fictionalist reading of "there
exists". But on an objectual reading, yes.
> 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?)
How about Goldbach's Conjecture?
> 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.
Or, perhaps, you could deny that a proper understanding of the universal
quantifier (taken as ranging over that set) guarantees determinacy of
truth-value for every universally quantified statement.
> ... 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.
...and provided also that one can lay claim to the appropriately classical
understanding of the quantifiers.
> 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.
That's a really nice way of putting it, modulo the rider about the
meanings of the quantifiers.
> 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").
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