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

**Date:** Wed Aug 18 2004 - 22:19:29 MDT

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--- Eliezer Yudkowsky <sentience@pobox.com> wrote:

*> Marc Geddes wrote:
*

*> >
*

*> > 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.
*

*>
*

*> But what makes you think the new axioms are correct?
*

*> If you add an axiom
*

*> to PA stating that PA is consistent - call this new
*

*> system PA+1 - what
*

*> makes you think that PA really *is* consistent?
*

*> This is what people
*

*> misunderstand about Godel's Theorem; they think we
*

*> know a truth that PA
*

*> doesn't. Actually, we just guess it.
*

We cannot of course, know with certainty that PA is

consistent. But we're not 'just guessing' either. We

simply treat mathematics like any other science: Give

up the unattainable dream of certainty and use

non-axiomatic reasoning. Perform 'mathematical

experiments' using computers and investigate

properties of various maths things, formulate

hypotheses, etc. i.e simply use Bayesian reasoning to

assign probabilities to any 'Godel undeciable'

statements.

All the Godel Theorem really proves is that we can't

attain 100% certainty. But this simply isn't the

terrible limitation that mathematicians keep claiming

it is. We can't attain 100% certainty in the physical

sciences either... but scientists don't run around

complaining that questions about science are

'undecidable'.

Here's comment from Greg Chaitin:

"Let me put it this way: Yes, I agree, mathematics and

physics are different, but perhaps they are not as

different as most people think, perhaps it's a

continuum of possibilities. At one end, rigorous

proofs, at the other end, heuristic plausibility

arguments, with absolute certainty as an unattainable

limit point."

A paper of his:

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/dijon.html

*>
*

*> > 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.
*

*>
*

*> Given that PA is consistent, I think that PA+1 has
*

*> the same model as PA...
*

*> actually, I don't know this, but it seems
*

*> "intuitively likely" and we all
*

*> know how reliable that is.
*

*>
*

*> The behaviors of physics are not analogous to
*

*> sentences in predicate
*

*> calculus since no real-world physical behavior
*

*> contains a universal or
*

*> existential quantifier. Physics doesn't need an
*

*> infinitely ascending
*

*> series of axioms to specify all finite physical
*

*> behaviors. The
*

*> unspecifiable axiom set would only be needed to
*

*> compress our description of
*

*> some infinite sets of finite physical behaviors into
*

*> a universally
*

*> quantified sentence asserted by the axiomatic
*

*> system.
*

PHYSICS doesn't need an infinitely ascending series of

axioms, but I think that SCIENCE in general does.

Consider the science of complex systems: New

'emergent' properties keep appearing that are not

explainable in terms of lower-term properties. For

instance the concept of 'Entropy' is probably an

emergent property not really explicable in terms of

the motions of individual particles (if you knew where

all the individual particles were the entropy of the

system would be 0).

*>
*

*> Let G(x) mean "x encodes a proof of the Godel
*

*> sentence in PA". The physics
*

*> of the Peano system suffice to specify the finite
*

*> behaviors ~G(1), ~G(2),
*

*> ~G(3). If PA is consistent, the physics suffice to
*

*> show ~G(x) for any
*

*> given number x. But manipulating the axiom system
*

*> will not enable you to
*

*> deduce, by logical manipulation of the axioms, that
*

*> Ax:~G(x) - you have to
*

*> check each x piecemeal, independently. Logical
*

*> deduction on the axioms
*

*> will enable you to conclude that Ax:~G(x) iff
*

*> Ax:~C(x) where C(x) means
*

*> that x encodes a proof of a contradiction in PA.
*

*> But logical deduction
*

*> will not let you deduce Ax:~G(x). Nor is this
*

*> something that I, or any
*

*> human mathematician, knows; we only guess it because
*

*> PA has worked pretty
*

*> well so far.
*

*>
*

*> The problem with Peano arithmetic is not that PA
*

*> doesn't completely
*

*> describe all finite numeric behaviors, but that PA
*

*> describes numeric
*

*> behaviors with less compressibility than we would
*

*> prefer - we can't
*

*> universally quantify some predicates that are
*

*> universally true (if PA is
*

*> consistent).
*

*>
*

*> --
*

*> Eliezer S. Yudkowsky
*

*> http://intelligence.org/
*

*> Research Fellow, Singularity Institute for
*

*> Artificial Intelligence
*

*>
*

*>
*

*>
*

=====

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- Gen. John Stark

"The Universe...or nothing!"

-H.G.Wells

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