[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

Quoting joeshipman at aol.com:


> I am rather 
> surprised at the number of replies that seem to feel Frege, Russell,  
> etc. did not actually accomplish anything (I am not claiming they fully 
> reduced mathematics to logic, just that a fragment equivalent to PA was 
> so reduced without needing the more powerful axioms of less clear 
> "logical" status that they needed for the rest of mathematics.)

Of course no reasonable person thinks that Frege and Russell did not
actually accomplish anything. They founded modern logic. But it is widely
thought that they failed to reduce mathematics, or even arithmetic, to logic
- in any normal sense of "logic".  Russell had to add the extra assumption
that there are infinitely many individuals in the lowest level, and he never
claimed that that is a truth of logic. 

> Raatikainen points out that with EmptySet, Extensionality, and the 
> capacity to adjoin an element to a set, one can already interpret 
> Robinson's Q; but in doing this he makes my point for me. I have been 
> saying all along that these and similar operations are so fundamental 
> and independent of "subject matter" that they deserve to be called 
> "logical", even if they require extending the predicate calculus 
> slightly. 

They issue then seems to become mostly verbal. But for me, like for many
others, logic concerns principally logically valid reasoning, and logical
truth. And from that perspective, such axioms do not count as logically
true.     
 
> Going from Q to PA here is nontrivial; I believe Friedman has a similar 
> interpretability result for PA but I don't recall its exact form. 

There is at least the old results of Brown & Wang (1966), where one adds to
the above three axioms a form of induction scheme, and gets a system in
which PA can be interpreted, and vice versa. This theory proves the axioms
of replacement, powerset, foundation and choice. (but, of course, not infinity).

 
> Raatikainen objects to my explication of logical validity as "true in 
> all models" -- I did not mean this in the precise sense he interprets 
> it, but I won't try to repair that "definition" here since we are 
> really arguing about what should count as "logical".

Yes, I take it literally. I think it is good to be precise here. 


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