[FOM] n-th order ZFC
Robert Black
mongre at gmx.de
Wed Jul 13 06:33:08 EDT 2011
Am 12.07.11 22:18, schrieb Roger Bishop Jones:
> It does not require any metaphysical position at all. Acceptance that
> "all possible subsets" is meaningful, like acceptance that the concept
> of natural number is meaningful, is well within the normal standards
> of mathematics, and is independent of metaphysical ontology. (it is
> true that philosophers do sometimes refer to the acceptance of the
> objectivity of truth in some domain as realism, but then some
> philosophers are unable to accept that one can believe in the
> objective truth of arithmetic without also believing in the existence
> of numbers. On that latter point I can assure them as a matter of
> empirical fact that it is possible).
I don't think you can escape philosophy quite so easily. It's true that
the view that mathematical sentences have truth values independently of
whether or not we can discover them (call this 'realism') is distinct
from the view that the abstract objects mathematical sentences appear to
talk about really do exist (call this 'platonism'), and neither position
trivially entails the other. But the fact that you can believe
arithmetical truths without *thinking that* you are thereby committed to
the existence of numbers hardly (yet) shows that you can believe
arithmetical truths without thereby *being* committed to the existence
of numbers.
As for the normal standards of mathematics, let's take the reals as an
example. It's a theorem of standard mathematics that there is up to
isomorphism only one complete ordered field. I think most mathematicians
unconcerned with foundational issues naively (but in my view also
correctly) take this theorem to mean just what it says: 'the reals' form
a completely determinate structure. But there's a different view which
doesn't of course deny the theorem, but regards it as always relativized
to a model of first-order set theory, as saying that *for any given
model of first-order set theory* all the complete ordered fields in that
model are isomorphic, but that the complete ordered fields in different
models need not be isomorphic to one another (also true of course). This
can then be combined with a denial that there is any such thing as the
intended model of (the first few transfinite ranks of) set theory to
reach a position where the original result is understood in a much
weaker way. I should add that there are philosophically and
mathematically highly skilled people on both sides of this divide: it
annoys me intensely when people think it's just *obvious* that one side
is right and the other wrong (not that I'm accusing Roger of that).
Robert
--
Robert Black
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