Date: Tue May 09 2006 - 22:36:27 MDT
Eli Yudkowsky wrote :
Nonetheless human engineers build CPUs that carry out multiplication, by
working in the restricted space of CPU designs that the chip engineer
*knows* carries out multiplication, which is a subset of all possible
chip designs that carry out multiplication. To build a bridge that
stays up, it is not necessary to be able to calculate of any possible
bridge whether it stays up. It is enough to know at least one bridge
design such that you can calculate that it supports at least 30 tons.
With regard to Godel's (first) Incompleteness Theorem - and the consequent
problem with infinite recursives, I think the bridge analogy is particularly
apt. (Why do people always mention bridge-building in connection with Godel's
Ely is entirely correct to say that it is only sufficient to know that the
bridge will, in practice, fulfil the purpose for which it is designed - i.e.
bear a load of 30 tons minimum.
However, in terms of engineering, it is possible to consider several
different bridge architectures - Cantilever, Suspension or Box-girder, for instance.
All 3 bridge designs may meet the 30 ton specification, which now introduces
real-world practical & engineering constraints, such as cost-effectiveness,
availability of materials, planning & time deadlines, etc., etc.
For those not familiar, (in a nutshell) -
Kurt Godel's First Incompleteness Theorem (1931) states that :
"For any formal logical system S, there will be a statement P of the
language of S, such that if S is consistent, neither P nor its negation can be
proved within S."
That little sentence ruined a lifetime's mathematical work of Russell &
Whitehead. But the lesson for us here, (to continue with the bridge analogy) is
that - if you spend too much time designing all possible bridges, you will find
that your competitors will have gone a few miles up or down-river, designed,
erected and opened their bridge to traffic - and they're already collecting
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