On 2/19/2011 3:39 PM, benjayk wrote:
Bruno Marchal wrote:
Isn't it enough to say everything that we *could* describe
in mathematics exists "in platonia"?
The problem is that we can describe much more things than the one we
are able to show consistent, so if you allow what we could describe
you take too much. If you define Platonia by all consistent things,
you get something inconsistent due to paradox similar to Russell
paradox or St-Thomas paradox with omniscience and omnipotence.
Why can inconsistent descriptions not refer to an existing object?
Because an inconsistent description implies everything, whether the
object described exists or not. From "Sherlock Holmes is a detective
and is not a detective." anything at all follows.
The easy way is to assume inconsistent descriptions are merely an arbitrary
combination of symbols that fail to describe something in particular and
thus have only the "content" that every utterance has by virtue of being
uttered: There exists ... (something).
But we need utterances that *don't* entail existence. So we can say
things like, "Sherlock Holmes lived at 10 Baker Street" are true, even
though Sherlock Holmes never existed.
So they don't add anything to platonia because they merely assert the
existence of existence, which leaves platonia as described by consistent
I think the paradox is a linguistic paradox and it poses really no problem.
Ultimately all descriptions refer to an existing object, but some are too
broad or "explosive" or vague to be of any (formal) use.
I may describe a system that is equal to standard arithmetics but also has
1=2 as an axiom. This makes it useless practically (or so I guess...) but it
may still be interpreted in a way that it makes sense. 1=2 may mean that
there is 1 object that is 2 two objects, so it simply asserts the existence
of the one number "two". 3=7 may mean that there are 3 objects that are 7
objects which might be interpreted as aserting the existence of (for
example) 7*1, 7*2 and 7*3.
The problem is not that there is no possible true interpretation of 1=2;
the problem is that in standard logic a falsity allows you to prove
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