On 21 Feb 2011, at 17:34, benjayk wrote:
Bruno Marchal wrote:
On 20 Feb 2011, at 13:13, benjayk wrote:
Brent Meeker-2 wrote:
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.
I think it is perfectly fine when something implies everything. For
me it
makes very much sense to think of everything as everything existing.
The distinction something existant / something non-existant is a
relative
one, in the absolute sense existence is all there is - and it
includes
relative non-existence (for example Santa Claus exists, but has
relative
non-existence in the set of things that manifests in a consistent
and
predictable way to many observers).
Aso, it emerges naturally from seemingly consistent logic that
everything
exists (see Curry's paradox).
Curry "paradox" was a real contradiction, Curry put his theory in the
trash the day he sees the contradiction, and begun some other less
ambitious theory (the illetive theory of combinators).
OK, but this doesn't change the rest of the rest of the argument.
Also, the Curry paradox is still there in natural language, which
seems
capable of making useful statements even though the Curry paradox
entails
the truth of every statement in natural language.
Natural language are very complex, and that is why we constraint the
machine to use formal language in the ideal case.
But even for natural language, it is usually accept that not all
sentence are true, and some fuzzy version of Tarski theory of truth
can already be helpful for many situation. In particular "snow is
white" is true because it is the case that snow is white.
Bruno Marchal wrote:
Brent Meeker-2 wrote:
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.
If we find something that doesn't entail existence, it still entails
existence because every utterance is proof that existence IS.
We need only utterances that entail relative non-existence or that
don't
entail existence in a particular way in a particular context.
You need some non relative absolute base to define relative
existence.
The absolute base is the undeniable reality of there being experience.
But this one is not communicable. It does play a role in comp, though.
But it is not enough. usually people agree with the axiom of Peano
Arithmetic, or the initial part of some set theory.
Bruno Marchal wrote:
Brent Meeker-2 wrote:
So we can say
things like, "Sherlock Holmes lived at 10 Baker Street" are true,
even
though Sherlock Holmes never existed.
Whether Sherlock Holmes existed is not a trivial question. He didn't
exist
like me and you, but he did exist as an idea.
Even if you met *a* Sherlock Holmes in Platonia, you have no cirteria
to say it is the usual fictive person created by Conan Doyle,
because,
in Platonia, he is not created by Conan Doyle, ...
In Platonia he is not created by Conan Doyle, which makes sense,
given the
possible that other people use the same fictional character, so he is
essentially discovered, not created.
But I don't know what you want to imply with that.
Just that fictionism, the idea that numbers are fiction of the same
type as fictive personage from novels does not make sense, except to
confuse matter.
Bruno Marchal wrote:
Brent Meeker-2 wrote:
So they don't add anything to platonia because they merely assert
the
existence of existence, which leaves platonia as described by
consistent
theories.
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
anything.
Yes, so we can prove anything. This simply begs the question what
the
anything is. All sentences we derive from the inconsistency would
mean the
same (even though we don't know what exactly it is).
We could just write "1=1" instead and we would have expressed the
same, but
in a way that is easier to make sense of.
This is not problematic, it only makes the proofs in the inconsisten
system
worthless (at least in a formal context were we assume classical
logic).
And it would make Platonia worthless. The "real", genuine, Platonia
is
already close to be worthless due to the consistency of inconsistency
for machine. This already put quite a mess in Platonia. By allowing
complete contradiction, you make it a trivial object.
Why? When we contradict ourselves we may simply interpret this as a
expression of the trivial truth of existence. This doesn't change
Plantonia
at all, because it exists either way.
The whole point of Gödel's theorem is that M proves 0=1 is different
from M proves provable('0=1'). The first implies the second, but the
second does not implies the first. The difference between G and G*
comes from this fact.
And why is inconsistency allowed for machine, but disallowed for other
objects?
Because if a machine proves "0=1", she will be in trouble, but if God
or Platonia proves "0=1", then we are *all* in trouble.
Bruno
http://iridia.ulb.ac.be/~marchal/
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