On 12 Sep 2012, at 15:28, Platonist Guitar Cowboy wrote:
On Wed, Sep 12, 2012 at 2:05 PM, benjayk <benjamin.jaku...@googlemail.com
> 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"?
The idea is that you can understand what they prove as much as you
understand what they assume, and this independently of what is the
understanding.
If *you* agree with the elementary axioms, and inference rule, then
you agree, or show a flaw, with the deduction presented to you.
The actual interpretation or belief (or disbelief), in the axiom is
private and the scientist is mute on this.
A scientist will never say "I know", in its field of competence, or
even outside (but for some reason that is rare: very often scientist
forget the scientific attitude in the field of colleagues, apparently).
Bruno
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?
The consequences of differing leaps of faith on axioms and
ontological bets shouldn't be taboo, if scientific search is to
remain sincere somehow, why restrict ourselves to the habitual ones?
Fruitful discussion from both of you, so thanks for sharing.
m
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