On 23 Feb 2011, at 17:37, benjayk wrote:



Bruno Marchal wrote:





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 we can say "there is an undeniable reality of there being experience". Isn't this communicating that there is the undeniable reality of there being
experience?

OK. I was using communicating in the sense of a provable communication. You cannot convince someone that you are conscious. If he decides that you are a zombie, you might better run, probably, but there is no way you could prove the contrary.




We merely communicate something that everbody already fundamentally knows.

That is correct also, I think.



Though some like to deny what they already know.


That is bad faith, and is common.





Bruno Marchal wrote:

But it is not enough. usually people agree with the axiom of Peano
Arithmetic, or the initial part of some set theory.
But Peano Arithmetics is not a non relative absolute base. It is relative to the meaning we give it and to the existence of some reality. 1+1=2 can have infinite meanings, that all are relative to our interpretation ("If I lay another apple into the bowl with one apple in it there are two apples" is
one of them) and there being meaning in the first place.

Hmm... Most people agrees on a standard meaning for the natural numbers, like in the Fermat theorem, or any theorem or conjecture in number theory, or when you are using numbers in computer science. 1+1 = 2 is true in all those interpretations, even if computer science we use also some algebra where 1+1=0. That does not contradict that the standard integer are all different from 0, except 0.






Bruno Marchal wrote:




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.
Well I didn't want to imply that. Fictionage personage usually refer to some relative manifestation of an idea, while numbers are a more general and
abstract notion.
And if they are fiction, they are very prevalent fiction (not just among
people but among nature), which makes them basically non-fiction.

OK.





Bruno Marchal wrote:



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.
If we know that something can be proven, how is it different from taking it
to be proven?

By incompleteness "provable(false) -> false" is not provable in the system.
I is true for the machine, but the machine cannot prove it.



The only difference I could see could be that "M proves
provable('0=1')" means "provable in another system".

No, by Gödel's construction it really means "provable by the system itself". It is still third person reference, and the machine is not necessarily aware that "she is that system". Later we can see she can *never* be aware of that, but she might trust her doctor, and make bet, etc.






Bruno Marchal wrote:


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.
I thought we already established that 0=1 can have a clear meaning
(equivalent to statements of the form 0*A+B=1*C+D in standard arithmetics),
and so it poses no problem.

?
I have no clue what you are saying here. If "0 = 1" means "I love chocolate", then of course "0=1" might be true. Again, we use the standard meaning. Natural numbers and finite sets and finite strings have a common interpretation rich enough so that we can agree on a little but powerful set of axioms. This is already less clear for the notion of sets, although most mathematicians believe so. But for the natural numbers, we do agree on those axioms, and their correspond to what has been taught at school.

If some my student defend ideas like 0 = 1, I give them a 0/10, and I reassure them that it is a good note because if 0 = 1, then 10 = 0. 10 = 1+1+1+1+1+1+1+1+1+1 = 0+0+0+0+0+ 0+0+0+0+0 = 0.



My suggestion is that every statement has such an interpretation. Circles with edges makes sense if we allow hyperreal numbers as numbers of edges and lenght of edges, triangles with four sides may mean such a geometric object: http://commons.wikimedia.org/wiki/File:Triangle-square-area-dev.png and that
God is omnipotent may mean anything.

Logic has been invented for avoiding interpretations as much as possible, and then for studying mathematically what can be interpretations, and the relations (Galois connection) between formal deduction and relations on interpretations. We force the "propositions" which mean anything to be eliminated, for helping the progress toward genuine truth and meaning.

We can say that first order logic does succeed in the interpretation elimination, thanks to a theorem of completeness (not incompleteness) by Gödel. A formula is a theorem IF and ONLY IF the formula is true in all interpretations.

Bruno

http://iridia.ulb.ac.be/~marchal/



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