[FOM] Falsify Platonism?

[Timothy Y. Chow]

On Sun, 25 Apr 2010, rgheck wrote:
> A good deal of care is needed here about what one means when one says 
> that the natural numbers are "defined" by this, that, or the other 
> thing. Personally, I'd prefer to talk of how our *concept* of the 
> natural numbers is defined; the numbers themselves are not defined any 
> more than I am. What a contradiction in PA would show, therefore, is 
> that there is something incoherent about our *concept* of the natural 
> numbers or, probably better, in our *concept* of the finite, 
> since---unless Q is inconsistent again---it is finitude that is really 
> at issue here. But it is unclear why such a failure should cast any 
> special light on Platonism. As has already been pointed out, if PA is 
> inconsistent, so is HA, which makes it all the more likely that the 
> issue is our *concept* of the finite.

This is precisely the kind of proposal that philosophers might find 
satisfying but that is not likely to satisfy the typical mathematician.  
If everything hinges on drawing a sharp distinction between the natural 
numbers and our concept of the natural numbers, then the mathematician is 
going to be perplexed, since this is not the kind of sharp distinction 
that is customarily demanded in mathematical discourse.  After all, which 
mathematical properties are satisfied by the natural numbers but not by 
our concept of the natural numbers, or vice versa?  Are they isomorphic or 
not?  If they're isomorphic, or if asking that question is a category 
error, then why should the mathematician care about the distinction?

I'll put the point another way.  If, in order to salvage platonism in the 
face of a contradiction in PA, one must sharply distinguish between N and 
the concept of N, then I think a contradiction in PA does indeed falsify 
the kind of (naive?) platonism that is common among practicing 
mathematicians, even if it leaves a subtler variety of platonism 
unscathed.

Tim