On 28 Aug 2009, at 10:47, marc.geddes wrote:

>
>
>
> On Aug 27, 7:35 pm, Bruno Marchal <marc...@ulb.ac.be> wrote:
>
>>
>> Zermelo Fraenkel theory has full transfinite induction power, but is
>> still limited by Gödel's incompleteness. What Gentzen showed is that
>> you can prove the consistency of ARITHMETIC by a transfinite  
>> induction
>> up to epsilon_0. This shows only that transfinite induction up to
>> epsilon_0 cannot be done in arithmetic.
>
> Yes.  That's all I need for the purposes of my criticism of Bayes.
> SInce ZF theory has full transfinite induction power, it is more
> powerful than arithmetic.
>
> The analogy I was suggesting was:
>
> Arithmetic = Bayesian Inference
> Set Theory =Analogical Reasoning


This makes no sense for me.

Also, here arithmetic = Peano Arithmetic (the machine, or the formal  
system).

Obviously (?, by Gödel) Arithmetic (arithmetical truth) is infinitely  
larger that what you can prove in ZF theory.

Of course ZF proves much more arithmetical true statements than PA.
Interestingly enough, ZF and ZFC proves the same arithmetical truth.   
(ZFC = ZF + axiom of choice);
And of course ZFK (ZF + existence of inaccessible cardinals) proves  
much more arithmetical statements than ZF.
But all those theories proves only a tiny part of Arithmetical truth,  
which escapes all axiomatizable theories.


>
> If the above match-up is valid, from the above (Set/Category more
> powerful than Arithmetic), it follows that analogical reasoning is
> more powerful than Bayesian Inference, and Bayes cannot be the
> foundation of rationality as many logicians claim.
>
> The above match-up is justified by (Brown, Porter), who shows that
> there's a close match-up between analogical reasoning and Category
> Theory.  See:
>
> ‘"Category Theory: an abstract setting for analogy and
> comparison" (Brown, Porter)
>
> http://www.maths.bangor.ac.uk/research/ftp/cathom/05_10.pdf
>
> ‘Comparison’ and ‘Analogy’ are fundamental aspects of knowledge
> acquisition.
> We argue that one of the reasons for the usefulness and importance
> of Category Theory is that it gives an abstract mathematical setting
> for analogy and comparison, allowing an analysis of the process of
> abstracting
> and relating new concepts.’
>
> This shows that analogical reasoning is the deepest possible form of
> reasoning, and goes beyond Bayes.


I agree, but there are many things going beyond Bayes.



>
>
>> I agree with your critics on Bayesianism, because it is a good tool
>> but not a panacea, and it does not work for the sort of credibility
>> measure we need in artificial intelligence.
>
> The problem of priors in Bayesian inference is devastating.  Simple
> priors only work for simple problems, and complexity priors are
> uncomputable.  The deeper problem  of different models cannot be
> solved by Bayesian inference at all:


Like all theorems, Bayes theorems can be used with many benefits on  
some problems, and can generate total non sense when misapplied.

Bruno

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




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