[FOM] Semantical realism without ontological realism in mathematics

[Neil Tennant]

On Wed, 21 May 2003, Aatu Koskensilta wrote:

> 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.

Let's call this position, defined in terms of bivalence, semantic realism
(SR).
 
> 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".

Let's call the view that the abstract objects of mathematics exist as
mind-independent objects, 'ontological realism' (OR).

I believe that each of the following four positions has adherents:

1.	SR + OR
2.	SR + not-OR
3.	not-SR + OR
4.	not-SR + not-OR

For (1), I would cite Frege.
For (2), I would cite Field.
For (3), I offer myself (well, not in the sense that I would *die* for the
position ...)
For (4), I would cite Brouwer.

Neil Tennant