[FOM] Falsify Platonism?
Markus.Pantsar at helsinki.fi
Mon Apr 26 08:02:44 EDT 2010
> 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?
Mathematician may not care about the distinction in his work, but if
he believes in the platonist existence of natural numbers, there
remains the possibility that PA ultimately does not capture these
numbers. So if one derives an inconsistency in PA, another
axiomatization of arithmetic could still do the trick. Of course most
of us have no problem in accepting that PA captures the concept of
natural number, but an inconsistency in PA would hardly be enough to
*falsify* platonism. Perhaps the questions you ask cannot be answered,
but that doesn't mean it is not *possible* that the number structure
of PA is not isomorphic to the platonist natural numbers (whatever
they may be).
I believe that platonism is way too elusive a philosophical position
to be falsified by mathematical proofs. The committed platonist will
find a way to defend his position no matter what. Against
contradictions, all he needs is a difference between the the world of
mathematical objects and our knowledge of that world - the possibility
of which is the very creed of platonism.
All the best,
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