On 30 Aug 2009, at 07:00, marc.geddes wrote:

>
>
>
> On Aug 30, 12:10 am, Bruno Marchal <marc...@ulb.ac.be> 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’


This does not make sense.




>
>
> 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’

This can make sense. Analogies are then seen as a generalization of  
morphism, which is the key notion of category theory.



>
>
>
>
>>
>>> 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.

You may develop. My feeling is that to compare category theory and  
Bayesian inference, is like comparing astronomy and fishing. They  
serve different purposes. Do you know Dempster Shafer theory of  
evidence? This seems to me addressing aptly the weakness of Bayesian  
inference.

Bruno

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




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