[FOM] Semantical realism without ontological realism in mathematics

[Aatu Koskensilta]

praatika at mappi.helsinki.fi wrote:
> The difference between "semantical realism" and "ontological realism" was 
> expressed quite clearly also by Dummett in his "Realism and anti-realism", 
> in his The Seas of Language, O.U.P, 1993.
> 
> He says that it is possible to be e.g. a neo-Fregean platonist about 
> mathematical objects but nevertheless deny that they have any properties 
> other than those we are capable of recognizing, and that it is also 
> possible to be a Dedekindian who maintain that mathematical objects are 
> free creations of human mind but may nevertheless have, once created, 
> properties independently of our capacity to recognize them.    

I'm aware of these possibilities. However, my interest lies in whether 
there are any explicitly definable or explicable realist semantics for 
higher mathematics that doesn't require ontological realism in any 
sense, i.e. which does not depend on the notion of existence of 
mathematical objects, whichever form this existence is supposed to take 
(being mental creations, residing in the platonistic realm, ...), or in 
which the notion of existence is explicitly reduced to some non-realist 
notions.

For example, the Dedkindian position that mathematical objects are free 
creations of the human mind is all fine and dandy. What I'm interested 
in is not the exact mode of "subsistence" taken for substitute for 
platonistic existence, but the semantics; it is not to me at all obvious 
  how we can have realist semantics for classical mathematics (as 
opposed to, say, constructive mathematics) when our mathematical 
ontology consists of our free creations. If Dedekind's position is taken 
literally, I don't even see how it could be said that there is an 
infinite number of mathematical objects (which is why the axiom of 
infinity was adopted by Russell), even less how there could be an 
uncountable number of mathematical objects.

But perhaps I'm being misled by a similar confusion as that with regards 
to the constructive/platonistic notion of uncountability. If 
uncountability of the universe of our mathematical objects simply means 
the non-existence of a freely created bijection between them and the 
freely created naturals, it could very well turn out to be true.

If, again, we take the Tarskian definition for truth and substitute for 
"exists" the expression "is a free creation of human mind" I don't see 
how we could get a realist semantics, unless we interprete "free 
creation of human mind" so loosely it becomes equivalent to "exists in 
an ideal mental platonist realm", as apparently Dedekind did.

For Roger Bishop Jones and for Carnap all these questions are 
meaningless, or are in fact merely questions about the framework of the 
metalanguage. They may very well be right. However, the question can be 
rephrased so as to satisfy their requirements -- although perhaps they 
would find the question to be of little or no interest -- by asking 
whether there exists a model (in the sense that Tarskian semantics as 
usually formulated may be thought to be a model of the platonistic 
conception of semantics, not in the sense of model theory) of meaning 
and truth in mathematics which *explicitly* does not presuppose the 
existence of abundant supply of arbitrarily substantial mathematical 
objects.

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

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