On Aug 30, 12:10 am, Bruno Marchal <marc....domain.name.hidden> wrote:
>
> >http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
>
> > "The TRUE but unprovable statement referred to by the theorem is often
> > referred to as “the Gödel sentence” for the theory. "
>
> > The sentence is unprovable within the system but TRUE. How do we know
> > it is true? Mathematical intuition.
>
> Not really. The process of finding out its own Gödel sentence is
> mechanical. Machines can guess or infer their own consistency, for
> example. In AUDA intuition appears with the modality having "& p" in
> the definition (Bp & p, Bp & Dp & p).
> Those can be related with Bergsonian time, intuitionistic logic,
> Plotinus universal soul, and sensible matter.
>
>
>
> > So to find a math technique powerful enough to decide Godel
> > sentences ,
>
> This already exists. The diagonilization is constructive. Gödel's
> proof is constructive. That is what Penrose and Lucas are missing
> (notably).
> The truth of Gödel sentences are formally trivial. That is why
> consistency is a nice cousin of consciousness. It can be shown to be
> true easily by the system, and directly (in few steps), yet remains
> unprovable by the system, not unlike the fact that we can be quasi
> directly conscious, yet cannot prove it. Turing already exploited this
> in his "system of logic based on ordinal" (his thesis with Church).
>
Penrose deals with this point in ‘Shadows of The Mind’ (Section 2.6,
Q6);
‘although the procedure for obtaining (Godel sentences from a formal
system) can be put into the form of a computation, this computation is
not part of the procedures contained in (the formal system). It
cannot be, because (the formal system) is not capable of ascertaining
the truth of (Godel sentences), whereas the new computation – together
with (the formal system) is asserted to be able to’
In ‘I Am a Strange Loop’, Hofstadter argues that the procedure for the
determining the truth of Godel sentences is actually a form of
analogical reasoning. (Chapters 10-12)
(page 148)
‘by virtue of Godel’s subtle new code, which systematically mapped
strings of symbols onto numbers and vice versa, many formulas could be
read on a second level. The first level of meaning obtained via the
standard mapping, was always about numbers, just as Russell claimed,
but the second level of meaning, using Godel’s newly revealed mapping…
was about formulas’
…
(page 158)
‘all meaning is mapping mediated, which is to say, all meaning comes
from analogies’
>
> > Bayesian reasoning (related to) functions/relations
> > Analogical reasoning (related to) categories/sets
>
> Those are easily axiomatized.
> I see the relation "analogy-category", but sets and functions are
> together, and not analogical imo.
> I don't see at all the link between Bayes and functions/relations.
> Actually, function/relations are the arrows in a category.
See what I said in my first post this thread. The Bayes theorem is
the central formula for statistical inference. Statistics in effect
is about correlated variables. Functions/Relations are just the
abstract (ideal) version of this where the correlations are perfect
instead of fuzzy (functions/relations map the elements of two sets).
That’s why I say that Bayesian inference bears a strong ‘family
resemblance’ to functions/relations.
You agreed that analogies bear a strong ‘family resemblance’ to
categories.
Category theory *includes* the arrows. So if the arrows are the
functions and relations (which I argued bears a strong family
resemblance to Bayesian inference), and the categories (which you
agreed bear a family resemblance to analogies) are primary, then this
proves my point, Bayesian inferences are merely special cases of
analogies, confirming that analogical reasoning is primary.
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Received on Sat Aug 29 2009 - 22:00:41 PDT