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/
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to [email protected].
To unsubscribe from this group, send email to
[email protected].
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en.
