[FOM] Semantical realism without ontological realism in mathematics

[Aatu Koskensilta]

Roger Bishop Jones wrote:

 > 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.

And you wouldn't count yourself as an anti-realist either, I gather.

 > 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.

This is the Carnapian distinction between internal and external 
questions, I take it.

 > 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.

I can understand that. I dind't mean to actually offer it as
a plausible interpretation, I just wanted to point out that
in case of arithmetic, there's an explicit interpretation
or construction which preserves semantical realism without
assuming ontological realism (for mathematical
objects of some substance, e.g. infinite sets and such like).

Of course, if one thinks accepting the Tarskian definition of truth
(as it now stands) for some domain does not imply also endorsing
ontological realism for that domain, then the interpetation
I presented offers nothing of value. This seems to be your
position.

However, if one does accept Tarskian definition of truth --> ontological
realism, the question becomes whether there are any modifications of
the Tarskian definition which would preserve its realistic nature
while freeing its applications from substantial ontological commitments.

For example, consider the statement phi = Ex(x = the set of real 
numbers). Now according to the T-schema

  phi is true <--> Ex(x = the set of real numbers)

In other words, the sentence "the set of real numbers exists" is true if 
and only if the set of real numbers exists. Otherwise it is false. It 
would seem that unless we insist most of statements of mathematics
are simply false, there in fact exists a substantial number of 
mathematical objects.

One can get around this by claiming that the informal language does not
have Tarskian semantics. That is

  "the set of real numbers exists" is true <--> the set of real numbers
  exists

but on the meta-meta-level

  ""the set of real numbers exists" is true " is true <--> something
  else not equivalent to the set of real numbers existing

-- 
Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus