**From:** Eliezer Yudkowsky (*sentience@pobox.com*)

**Date:** Thu Aug 19 2004 - 04:15:09 MDT

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

Marc Geddes wrote:

*>>
*

*>> But what makes you think the new axioms are correct? If you add an
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*>> axiom to PA stating that PA is consistent - call this new system PA+1
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*>> - what makes you think that PA really *is* consistent? This is what
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*>> people misunderstand about Godel's Theorem; they think we know a truth
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*>> that PA doesn't. Actually, we just guess it.
*

*>
*

*> We cannot of course, know with certainty that PA is consistent. But
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*> we're not 'just guessing' either. We simply treat mathematics like any
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*> other science: Give up the unattainable dream of certainty and use
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*> non-axiomatic reasoning. Perform 'mathematical experiments' using
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*> computers and investigate properties of various maths things, formulate
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*> hypotheses, etc. i.e simply use Bayesian reasoning to assign
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*> probabilities to any 'Godel undeciable' statements.
*

Goodstein(4) returns to 0 after (3*2^402653211 - 3) steps. With a bit of

epistemological handwaving and questionable (i.e. wrong) priors you could

apply the Rule of Succession to arrive at a confidence, after one billion

steps, that the next *single* step was unlikely to halt, the confidence in

halting being 1/1,000,000,002 or less. But the Rule of Succession is

entirely *agnostic* about the confidence that Goodstein(4) *never* halts,

ever, even after a quadrillion observations - and in fact Goodstein(4) does

halt.

It may seem reasonable to say that since the Peano axioms have been

extensively tested and have never yet produced a contradiction, therefore

they probably never will. Actually, *all* the Rule of Succession gives you

is a high confidence that the next *particular* result published in some

math journal will not be a reductio of PA. But to claim that PA never

produces a contradiction *ever* is something wholly unjustified by the

evidence available to us, at least on the Rule of Succession.

A quintillion observations may seem like a lot of evidence, but to

generalize from there to infinity is unjustified. Practically all finite

numbers are larger than than those in your sample set. There are just too

many empirical observations in math that hold true for 3 -> 3 -> 64 -> 2

steps and then break at 3 -> 3 -> 65 -> 2. For example, the test: "Is X

less than Graham's number?"

Contemporary Bayesian math will let you reason from a huge mound of

empirical evidence to a high confidence in the next single case, but *not*

to an infinite case or a universal generalization - which, when you think

about it, is common sense. Come to think, I haven't seen this pointed out

anywhere, but it seems obvious enough.

Laplace originally used the Rule of Succession to derive that if the sun

has risen for 2,000,000 consecutive mornings, then the naive confidence of

the sun rising tomorrow is 2,000,001/2,000,002. But despite taking into

account six orders of magnitude more evidence (Laplace reasoned on the last

5000 years), my own confidence would be at least two orders of magnitude

lower (implied if the Sun seems at least 25% likely to be switched off in

the next 50 years), and we all know it's going to fail eventually. The

thing about a Bayesian confidence is that it's designed on the assumption

of being occasionally wrong.

-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence

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