FOM: Re: Ontology in Logic and Mathematics

[Roger Bishop Jones]

In response to Joe Shipman Friday, September 15, 2000 9:49 PM


> Roger Jones wrote:
>
> >my view is that neither logic nor mathematics need or can make any
> >absolute ontological commitments, and I doubt that a worthwhile
> >distinction between logic and mathematics can be founded on ontology.<<
>
> This seems correct, if you are talking about "absolute" ontological
> commitments.  But the kind of ontological commitments made by
> mathematics are different in kind.  Consider two specific theorems:
>
> 1) For all k, l, m there exists n so large that, if you color the
> k-element subsets of {1,...,n} with l colors, then there will be a
> subset X of cardinality at least m all of whose k-elements subsets have
> the same color, and such that the cardinality of X is greater than the
> smallest element of X. (Paris-Harrington)
>
> This is  provable in ZFC, but any such proof will use the Axiom of
> Infinity.  Does this mean you must be ontologically committed to the
> existence of an infinite set in order to accept this theorem as true?

No. (and similarly for (2))

You only have to accept that the axiom of infinity is true in the intended
domain of discourse, e.g. in all models of ZFC (if that is the intended
domain of discourse).
If you leave open the semantics of the language, then what you assert is
also left open.

Roger Jones