From: Eliezer Yudkowsky (sentience@pobox.com)
Date: Thu Aug 19 2004 - 04:15:09 MDT
Marc Geddes wrote:
>>
>> 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.
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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