On 12 Sep 2012, at 21:48, benjayk wrote:
Platonist Guitar Cowboy wrote:
On Wed, Sep 12, 2012 at 2:05 PM, benjayk
<[email protected]>wrote:
Bruno Marchal wrote:
On 11 Sep 2012, at 12:39, benjayk wrote:
Our discussion is going nowhere. You don't see my points and
assume
I want to
attack you (and thus are defensive and not open to my criticism),
and I am
obviously frustrated by that, which is not conducive to a good
discussion.
We are not opertaing on the same level. You argue using rational,
"precise"
arguments, while I am precisely showing how these don't settle
or even
adress the issue.
Like with Gödel, sure we can embed all the meta in arithmetic, but
then we
still need a super-meta (etc...).
I don't think so. We need the understanding of elementary
arithmetic,
no need of meta for that.
You might confuse the simple truth "1+1=2", and the complex truth
"Paul understood that 1+1=2". Those are very different, but with
comp,
both can be explained *entirely* in arithmetic. You have the
right to
be astonished, as this is not obvious at all, and rather counter-
intuitive.
There is no proof that can change this,
and thus it is pointless to study proofs regarding this issue (as
they just
introduce new metas because their proof is not written in
arithmetic).
But they are. I think sincerely that you miss Gödel's proof. There
will be opportunity I say more on this, here, or on the FOAR
list. It
is hard to sum up on few lines. May just buy the book by Davis (now
print by Dover) "The undecidable", it contains all original
papers by
Gödel, Post, Turing, Church, Kleene, and Rosser.
Sorry, but this shows that you miss my point. It is not about some
subtle
aspect of Gödel's proof, but about the main idea. And I think I
understand
the main idea quite well.
If Gödels proof was written purely in arithmetic, than it could
not be
unambigous, and thus not really a proof. The embedding is not
unique, and
thus by looking at the arithmetic alone you can't have a unambigous
proof.
Some embeddings that could be represented by this number relations
could
"prove" utter nonsense. For example, if you interpret 166568 to
mean "!="
or
"^6" instead of "=>", the whole proof is nonsense.
Thus Gödel's proof necessarily needs a meta-level, or
alternatively a
level-transcendent intelligence (I forgot that in my prior post)
to be
true,
because only then can we fix the meaning of the Gödel numbers.
You can, of course *believe* that the numbers really exists beyond
their
axioms and posses this transcendent intelligence, so that they
somehow
magically "know" what they are "really" representing. But this is
just a
belief and you can't show that this is true, nor take it to be
granted
that
others share this assumption.
Problem of pinning down "real representation" in itself aside. Can
"human"
prove to impartial observer that they "magically know what they are
really
representing" or "that they really understand"?
How would we prove this? Why should I take for granted that humans do
this,
other than legitimacy through naturalized social norms, which
really don't
have that great a track record?
Can we even literally prove anything apart from axiomatic systems at
all? I
don't think so. What would we base the claim that something really
is a
proof on?
The notion of proving seems to be a quite narrow and restricted one
to me.
That is why we have other notion than proof---which is of the type"
belief" (no "Bp -> p" in general), like knowledge, feeling,
experience, etc.
Incompleteness makes possible to recover by intensional nuances:
for a fixed machine B (I identify the machine with her beliefs) all
the Bp, Bp & p, Bp & Dt, Bp & Dt & p, etc. concerns exactly the same
arithmetical propositions, but obeys quite different logics
(classical, intuitionist, quantum-like, etc.).
Bruno
Apart from that, it seems human "understanding" is just delusion in
many
cases, and the rest is very limited at best. Especially when we
think we
really understand fundamental issues we are the most deluded.
benjayk
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