From: Marc Geddes (email@example.com)
Date: Wed Aug 18 2004 - 21:44:05 MDT
--- Christian Szegedy <firstname.lastname@example.org> wrote:
> Marc Geddes wrote:
> >All of those so-called 'uncomputable' maths
> >are in fact computable to any degree of accuracy
> >than 100% (so we can in fact compute the functions
> >with 95%, 99%, 99.9% or any degree of accuracy we
> >desire less than 100%)
> >Similairly, all of those so-called 'undecidable'
> >truths in maths are in fact decidable to any
> >confidence level less than 100% (so we could in
> >produce a non-axiomatic probabilistic argument to
> >achieve 95%, 99%, 99.9% or any degree of confidence
> >desire less than 100%)
> >Make sense?
> I would like to see your definition of "degree of
> accuracy" and
> "degree of confidence". I seriously doubt that you
> can define it in a
> sensible way so that you can arbitrarily approximate
> 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
(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
(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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