[FOM] Semantical realism without ontological realism in mathematics

[Roger Bishop Jones]

On Wednesday 21 May 2003  7:30 am, Aatu Koskensilta wrote:

> What sort of possibilities are there for semantical realism
> for set theory or other branches of mathematics *without*
> ontological realism?

I think I would count myself as a "realist" about the truth
of sentences in languages whose semantics is well-defined,
but not a realist concerning abstract ontology.

The philosopher whose position is closest to my own,
is Rudolph Carnap, and a good place to look for this
is his "Empiricism, Semantics and Ontology", which
appears as supplement A to "Meaning and Necessity".

I should mention perhaps that though Carnap and myself
are not realists in relation to abstract ontology,
we do not deny realism.  The question of realism is
considered to be without meaning, neither true nor
false.  If you ask us whether natural numbers exist,
we fail to understand your question, unless you ask
the question in some specific linguistic framework
which gives it meaning, e.g. in set theory, where
it turns out to be true.

Your attempt to justify a non-realist position by
giving a syntactic interpretation to abstract theories,
seems to me, and possibly would have to Carnap,
a kind of nominalism, and is quite unnecessary
from our point of view.

Roger Jones