**From:** Marc Geddes (*marc_geddes@yahoo.co.nz*)

**Date:** Wed Aug 18 2004 - 21:44:05 MDT

**Next message:**Marc Geddes: "Re: All is information (was: All is number)"**Previous message:**Thomas Buckner: "Re: All is information (was: All is number)"**In reply to:**Christian Szegedy: "Re: All is information"**Next in thread:**Christian Szegedy: "Re: All is information"**Reply:**Christian Szegedy: "Re: All is information"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

--- Christian Szegedy <szegedy@or.uni-bonn.de> wrote:

*> Marc Geddes wrote:
*

*>
*

*> >All of those so-called 'uncomputable' maths
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*> functions
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*> >are in fact computable to any degree of accuracy
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*> less
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*> >than 100% (so we can in fact compute the functions
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*> >with 95%, 99%, 99.9% or any degree of accuracy we
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*> >desire less than 100%)
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*> >
*

*> >Similairly, all of those so-called 'undecidable'
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*> >truths in maths are in fact decidable to any
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*> >confidence level less than 100% (so we could in
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*> fact
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*> >produce a non-axiomatic probabilistic argument to
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*> >achieve 95%, 99%, 99.9% or any degree of confidence
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*> we
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*> >desire less than 100%)
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*> >
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*> >Make sense?
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*> >
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*> >
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*> I would like to see your definition of "degree of
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*> accuracy" and
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*> "degree of confidence". I seriously doubt that you
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*> can define it in a
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*> sensible way so that you can arbitrarily approximate
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*> any
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*> uncomputable function or mathematical statements.
*

*>
*

Well let me turn that around and ask if there is any

reason why I SHOULDN'T be able to arbitrarily

approximate any 'uncomputable' functions or

'undecidable' maths statements?

What do the undecidability and uncomputability

theorems of people like Godel and Turing actually say:

In the case of uncomputability, that some functions

have infinite complexity. In the case of

undecidability, that the sea of maths truth is

infinite. What does this imply?

(a) No finite algorithim can achieve 100% accuracy in

specifying all maths functions

and

(b) No finite algorithim can achieve 100% certainty

about the turth status of all maths statements

That's it. That's all. But this is really not a

limitation at all in any PRACTICAL sense. Because:

(a) There is no reason why a finite algorithim

couldn't achieve a practical level of accuracy

arbitrarily close to 100% about any maths function

and

(b) There is no reason why a finite algorithim

couldn't a practical level of certainty arbitrarily

close to 100% about any maths statement

It's really ironic that mathematicians talk about

'uncomputable' maths functions and then they start

investigating them in detail using...COMPUTERS ;)

The Mandelbrot set for instance, is supposed to be

'uncomputable', yet there are lots of great pictures

generated by COMPUTERS representing the set perfectly

well to any desired degree of accuracy.

=====

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**Next message:**Marc Geddes: "Re: All is information (was: All is number)"**Previous message:**Thomas Buckner: "Re: All is information (was: All is number)"**In reply to:**Christian Szegedy: "Re: All is information"**Next in thread:**Christian Szegedy: "Re: All is information"**Reply:**Christian Szegedy: "Re: All is information"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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