From: Marc Geddes (marc_geddes@yahoo.co.nz)
Date: Wed Aug 18 2004 - 00:20:02 MDT
--- Eliezer Yudkowsky <sentience@pobox.com> wrote:
> Marc Geddes wrote:
> >
> > I don't see that an infinity of axioms (reality is
> > uncountable) is a problem. We are not limited to
> > axiomatic reasoning. Although higher level
> > mathematical axioms would not be derivable from
> lower
> > level mathetical axioms, we can still reason about
> and
> > *prove* the higher level axioms using
> probabilistic
> > reasoning.
>
> IANAM but that sounded to me like nonsense.
>
> --
> Eliezer S. Yudkowsky
> http://intelligence.org/
> Research Fellow, Singularity Institute for
> Artificial Intelligence
>
I don't think I was talking gibberish ;) I could
probably have worded it better but what I said still
seems O.K to me.
Let me try to rephrase:
Godel's theorem says that any system of finite axioms
(complex enough at least to incorporate both addition
and multipication) which is consistent, cannot be
complete. There will be some so-called 'undecidable'
truths, in the sense of perfectly sensible
propositions stated in the language of the system
which cannot be proved from within the system.
However are these truths really 'undecidable'? No,
not in any absolute sense. They would be perfectly
decidable from the perspective of a broader
mathematical system - one which contained the old
system of axioms plus some extra ones.
Now, if reality is a system of infinite axioms (which
cannot be finitely specified), then ALL mathematical
truths would in fact be decidable. Every mathematical
truth would be embedded in a broader axiomatic system
than the one needed to state it as a proposition.
All 'undecidable' actually means is that some
mathematical propositions cannot be proved with 100%
certainty. But there is nothing which says we can't
reason that the proposition is 70% likely to be true,
80% likely to be true, 90% likely to be true, or any
degree of certainty less than 100% (we simply deploy
non-axiomatic reasoning and treat mathematics like an
'experimental science' - using computers to perform
'experiments' on mathematical objects).
So you see, this whole notion of something being
'Godel undecidable' has been totally mis-interpreted
by the lay-man.
If a maths proposition is 'Godel undecidable' all that
means is that we can't deploy axiomatic reasoning and
achieve certainty regarding its truth status. But by
deploying non-axiomatic probabilistic reasoning, all
of those so-called 'undecidable' statements are in
fact decidable so long as we are prepared to accept
probabilities less than 100%
I repeat: the terms 'undecidable' and 'uncomputable'
have been totally misunderstood by mathematicians and
lay-men.
All of those so-called 'uncomputable' maths functions
are in fact computable to any degree of accuracy less
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 fact
produce a non-axiomatic probabilistic argument to
achieve 95%, 99%, 99.9% or any degree of confidence we
desire less than 100%)
Make sense?
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