[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

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

> It pains me to see this statement:
> 
>     I still think that even the axioms of successor are not 
>     really logically true, for they are false 
>     in all finite models.
> 
> It seems far more appropriate (and, yes, logical) to observe that there
> are no finite models of PA, and that is a logical consequence of the 
> axioms of successor.  I think we can be farely confident that Peano 
> intended that conseqence.

The "painful" quote was from me...

It is perfectly standard to explicate "logically true" as "true in all
models", such that any formula that is false in some model, e.g., in some
finite model, is not logically true. In this sense, the axioms of successor
are, quite trivially, not logically true. 

Obviously it is a fact of (mathematical) logic (as a field of study), more
exactly, of model theory, that PA has no finite models. But certainly this
is not logically true; proving it requires some non-logical axioms. 

Certainly Peano - or, Dedekind, who really invented the axioms - intended
the axioms to rule out finite models, but that is irrelevant for the present
issue. 

> I suspect that PA is of little utility for significant chunks of
> mathematics (i.e., real analysis and calculus),

Wrong. For example ACA_0, which is a conservative extension of PA, is
sufficient for practically all ordinary analysis or calculus. 

> while it is already too much for those
> logicians concerned about predicativity. 

Wrong. PA and ACA_0 are predicative. 

 
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