[FOM] Semantical realism without ontological realism in mathematics

[Aatu Koskensilta]

By mathematical semantical realism I basicly mean the position that 
mathematical statements are true or false independent of the 
mathematicians studying or knowing their truth or falsity. Also, realist 
position as regards to semantics insists that all statements obey the 
law of bivalence.

Now, if we take the position that Tarskian correspondence theory of 
truth is the only viable realistic theory of truth, it would seem that 
accepting semantical realism would force us to accept ontological 
realism in mathematics as well, for the (relevant portions of) Tarskian 
recursive definition of truth actually boil down to certain objects 
having the properties and relations the sentences ascribes them, and 
it's hard to see how any existential statement could be true in the 
Tarskain sense if "the mathematical objects don't exist out there".

Also, if we consider alternatives to ontological realism, say 
intuitionism, it seems that the concept of truth for the objects of 
these alternatives is not realistic (in the sense I use the term).

In the case of arithmetic we surely *can* do without ontological 
realism: simply replace the concept of intended model with "the set of" 
all true formulas without quantifiers and interprete the quantifiers 
substitionally. The set will include formulas such as "1+1=2 & 2+2=4", 
"~1=0" and so forth. This set is clearly recursive, and even though we 
might wish to not speak of it as a "set" (because our ontological 
qualms) this usage is no more harmful than speaking about the "set of 
sentences in English". Now, a formula phi is true iff

  1) phi \in true quantifierless formulas
  2) phi = \exists x chi and there is a statement which is a substitution
     instance of chi (i.e. chi with x replaced with a numeral) which is
     true
  3) phi = \forall x chi and all substitution instances of chi are true

If we treat arithmetic this way, we get a realistic conception of truth 
for arithmetic without postulating the existence of "natural numbers". 
In fact, we get something like "number theory as the general theory of 
elementary calculation or elementary numerical identities".

There are numerous objections which can be raised; for example, it could 
be argued that the formulation of this semantical picture requires 
already the concept of natural numbers. However, there is nothing that 
says that the truth *also* in the metalevel should not be understood 
according to this picture. Other objections are possible, but these are 
not really important for my purposes right now.

Now, as to set theory and other more abstract branches of mathematics, 
this clearly cannot be done; for us to do this with set theory we should 
begin with true statements of the form "a \in b", but this already 
presupposes that we have a name for every set. Obviously, the set of 
true statements of this form is a proper class, and every bit as 
porblematic as the proper class that is the universe of set theory.

Thus one might be led to the view that while one should (/it is possible 
to) accept semantical realism for arithmetic without ontological 
realism, this can't be done with regards to set theory.

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

(The reason I'm interested in this apart from the general philosophical 
importance is that I'm convinced that the concept of truth we use simply 
is realistic. However, it also seems clear to me that some property of a 
concept we made up -- in this case truth -- can give us absolutely no 
information about the world itself -- apart from the portions of world 
having to do with human concept formation, communication, &c -- and thus 
semantical realism should *not* imply ontological realism about any sort 
of entities.)

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

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