From: Anders Sandberg (asa@nada.kth.se)
Date: Mon Jun 17 2002 - 08:29:14 MDT
On Mon, Jun 17, 2002 at 09:21:41AM +1000, Colin Hales wrote:
>
> I seem to remember (1970's?) people started (more generally) doing
> mathematical
> proofs that consisted of pure numerical set exhaustion . The results of
> science became, in effect, a computer output. The general community
> when through a phase of discomfort - 'how can this be a real proof?'.
>
> The process challenged the long held reliance on separable, algorithmic
> a-priori equations as solutions. Somehow the computational proof seemed
> less valid, and yet no-one could mount an argument that, indeed,
> these were not proofs. A sqirming discomfort seemed to pervade the various
> dsciplines.
>
> Wolframs idea seems to be an extrapolation of this form of computational
> 'proof' and, I think, is saying that it is the only way that real
> progress will be made is to truly lose the remaining belief in the
> formulaic proof, where we see the formulaic proof as a special case
> in a more generalised computational regime, or perhaps a way of
> extracting general properties from a complex computational regime.
The problem with Wolfram's approach seems to be that he is not
really doing the exhaustive scans over problem domains, he is
rather using an inductive approach: show that a number of small
base cases produce certain behaviors (by exhaustive search), do
some estimates of more complex cases (by random sampling), and
then assume the pattern holds. It may be very convincing, but it
is like saying that "Li(x)-pi(x) has always been > 0 for all x I
have checked, so it is always so" - it may be false for a x larger
than the one I checked (in this case some x smaller than Skeve's
number). If the claim instead is "property X is generic" it is
even more convincing, but it doesn't really tell us what
conditions are necessary and sufficient for property X.
There are an infinite amount of CAs and similar systems that can
simulate fluid dynamics. Which of them are correct descriptions of
the world? The issue become very relevant when building a boat,
and the only solution Wolfram seems to give is to make a lot of
measurements and then remove the rules that won't fit - but you
still have an infinite amount of rules. Then you will likely use
Occham's razor/algorithmic complexity and try to find the simplest
remaining rule - but this is also enormously expensive. The other
approach, assume a basic physics of particles and construct a
description of fluid dynamics from their interactions seems to
require far less computation. Neither will reach Absolute
Certainty, but the goal is after all to get physics that we can
use when building boats or refridgerators.
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y
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