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From: marc.geddes <marc.geddes.domain.name.hidden>

Date: Sun, 30 Aug 2009 01:34:27 -0700 (PDT)

On Aug 30, 7:05 pm, Bruno Marchal <marc....domain.name.hidden> wrote:

*>
*

*> This does not make sense.
*

You said;

*>The truth of Gödel sentences are formally trivial.
*

*>The process of finding out its own Gödel sentence is
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mechanical.

*>The diagonilization is constructive. Gödel's
*

proof is constructive. That is what Penrose and Lucas are missing

(notably).

This contradicts Godel. The truth of any particular Godel sentence

cannot be formally determined from within the given particular formal

system - surely that's what Godel says?

The points are addressed in ‘Shadows of The Mind’ (Section 2.6,

Q6).

The point of Penrose/Lucs is that you can only formally determine the

Godel sentence of a given system from *outside* that system. We

cannot determine *our own* Godel sentences formally, and that's why we

have to rely on analogical reasoning (which is the argument of

Hofstadler in ‘I Am a Strange Loop’).

*>Analogies are then seen as a generalization of
*

morphism, which is the key notion of category theory.

Yes thats the sort of thing I'm suggesting, only I think its probably

the other way around, analogies are a particular type of morphism.

(morphism is more general)

*>You may develop. My feeling is that to compare category theory and
*

Bayesian inference, is like comparing astronomy and fishing. They

serve different purposes.

Well, Bayes is applied math, category theory is pure math. But its

all math. If category theory is the foundation of math, there must be

structures in there corresponding to Bayes.

*>
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*>
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*>
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*>
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*>
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*>
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*>
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*> > In ‘I Am a Strange Loop’, Hofstadter argues that the procedure for the
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*> > determining the truth of Godel sentences is actually a form of
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*> > analogical reasoning. (Chapters 10-12)
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*>
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*> > (page 148)
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*>
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*> > ‘by virtue of Godel’s subtle new code, which systematically mapped
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*> > strings of symbols onto numbers and vice versa, many formulas could be
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*> > read on a second level. The first level of meaning obtained via the
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*> > standard mapping, was always about numbers, just as Russell claimed,
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*> > but the second level of meaning, using Godel’s newly revealed mapping…
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*> > was about formulas’
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*> > …
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*> > (page 158)
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*>
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*> > ‘all meaning is mapping mediated, which is to say, all meaning comes
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*> > from analogies’
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*>
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*> This can make sense. Analogies are then seen as a generalization of
*

*> morphism, which is the key notion of category theory.
*

*>
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*>
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*>
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*>
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*>
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*>
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*>
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*> >>> Bayesian reasoning (related to) functions/relations
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*> >>> Analogical reasoning (related to) categories/sets
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*>
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*> >> Those are easily axiomatized.
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*> >> I see the relation "analogy-category", but sets and functions are
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*> >> together, and not analogical imo.
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*> >> I don't see at all the link between Bayes and functions/relations.
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*> >> Actually, function/relations are the arrows in a category.
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*>
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*> > See what I said in my first post this thread. The Bayes theorem is
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*> > the central formula for statistical inference. Statistics in effect
*

*> > is about correlated variables. Functions/Relations are just the
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*> > abstract (ideal) version of this where the correlations are perfect
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*> > instead of fuzzy (functions/relations map the elements of two sets).
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*> > That’s why I say that Bayesian inference bears a strong ‘family
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*> > resemblance’ to functions/relations.
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*>
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*> > You agreed that analogies bear a strong ‘family resemblance’ to
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*> > categories.
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*>
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*> > Category theory *includes* the arrows. So if the arrows are the
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*> > functions and relations (which I argued bears a strong family
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*> > resemblance to Bayesian inference), and the categories (which you
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*> > agreed bear a family resemblance to analogies) are primary, then this
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*> > proves my point, Bayesian inferences are merely special cases of
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*> > analogies, confirming that analogical reasoning is primary.
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*>
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*> 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
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*> evidence? This seems to me addressing aptly the weakness of Bayesian
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*> inference.
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*>
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*> Bruno
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*>
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*> http://iridia.ulb.ac.be/~marchal/- Hide quoted text -
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*>
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*> - Show quoted text -- Hide quoted text -
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*>
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*> - Show quoted text -- Hide quoted text -
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*>
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*> - Show quoted text -
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Received on Sun Aug 30 2009 - 01:34:27 PDT

Date: Sun, 30 Aug 2009 01:34:27 -0700 (PDT)

On Aug 30, 7:05 pm, Bruno Marchal <marc....domain.name.hidden> wrote:

You said;

mechanical.

proof is constructive. That is what Penrose and Lucas are missing

(notably).

This contradicts Godel. The truth of any particular Godel sentence

cannot be formally determined from within the given particular formal

system - surely that's what Godel says?

The points are addressed in ‘Shadows of The Mind’ (Section 2.6,

Q6).

The point of Penrose/Lucs is that you can only formally determine the

Godel sentence of a given system from *outside* that system. We

cannot determine *our own* Godel sentences formally, and that's why we

have to rely on analogical reasoning (which is the argument of

Hofstadler in ‘I Am a Strange Loop’).

morphism, which is the key notion of category theory.

Yes thats the sort of thing I'm suggesting, only I think its probably

the other way around, analogies are a particular type of morphism.

(morphism is more general)

Bayesian inference, is like comparing astronomy and fishing. They

serve different purposes.

Well, Bayes is applied math, category theory is pure math. But its

all math. If category theory is the foundation of math, there must be

structures in there corresponding to Bayes.

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Received on Sun Aug 30 2009 - 01:34:27 PDT

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