[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

Quoting "Dennis E. Hamilton" <dennis.hamilton at acm.org>:

> I understand that the absence of finite models for PA is a metatheorem,
> if you will.  Of course it is.  So is that a purely mathematical
> metatheorem, or is it indeed "logical?"

Well, not logical, if by "logical" we mean logically true (what else it
could mean?). As I said, you need extra-logical axioms in order to prove it.  

> I just don't see granting of existence to a finite model, which is
> certainly external to what PA provides, as somehow more logical than that.

Let me try again. For me, and I presuppose for many others, logic is
primarily about logical consequence, or entailment, and the validity of
arguments. The inference from A to B is logically valid  iff "A -> B" is
logically true (as I understand "logically true": true in all models; i.e.,
it has no counter-example).

Now if one would rule out finite models, many obviously non-valid inferences
would end up as valid.  

It is of course in part verbal question how one wants to understand
"logically true", but this is at least the standard understanding. 

Note: All this has nothing to do with PA.


> I'm also puzzled, now that Panu mentions it, how "logically true" is
> meant to help us with the question of how much of math is logic?  
> Does the math that is logic have to be math that is logically true 
> in this sense?  Is that the crux of logicism?


Yes.


Best, Panu




Panu Raatikainen

Academy Research Fellow, The Academy of Finland
Docent in Theoretical Philosophy, University of Helsinki

Department of Philosophy
P.O.Box 9
FIN-00014 University of Helsinki
Finland

e-mail: panu.raatikainen at helsinki.fi