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Article 1836 of comp.ai.philosophy:
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>From: weemba@libra.wistar.upenn.edu (Matthew P Wiener)
Newsgroups: comp.ai.philosophy
Subject: Re: Physical limits when programming neurons and minds
Message-ID: <58123@netnews.upenn.edu>
Date: 3 Dec 91 21:05:15 GMT
References: <57850@netnews.upenn.edu> <1991Dec2.005246.2168@morrow.stanford.edu>
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In-reply-to: dow@nova1.stanford.edu (Keith Dow)

In article <1991Dec2.005246.2168@morrow.stanford.edu>, dow@nova1 (Keith Dow) writes:
>>I repeat: there may be metamathematical obstructions to solving the
>>relevant physics, even if it's just Schroedinger's equation.  No amount
>>of talent can solve the unsolvable--except by changing the rules.
>
>I don't know what metamathematics is.

Metamathematics is the mathematics of doing mathematics.

Consider the problem of trisecting the angle in geometry.  For over
two thousand years, the goal was to find a way to do this using just
ruler and compass.  Then Gauss proved the following metageometric
theorem: it is impossible to trisect the general angle using ruler
and compass.

This century has seen an incredible amount of metamathematics.  A
chunk of it is considered required background for AI--even on just
a philosophical level.  Two basic theorems are Turing's about the
universality of a certain Turing machine, and Goedel's about the
essential incompleteness of Peano arithmetic.

If you haven't heard of this, you're missing out on all the fun, and
you simply have no way to comprehend what I'm referring to.  Let me
recommend, despite my personal distaste, Douglas Hofstadter's famous
GOEDEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID.

>				        Also your sentence which I quote
>below, is logicaly inconsistent.  

It's not a logical sentence.  It's a human sentence, meant for humans
to read and understand.  Like most examples of non-logical things that
AIs can't figure out, the human understanding requires a good deal of
background knowledge.  If you read it literally, you will fail to catch
on.  So I will translate....

>"No amount of talent can solve the unsolvable--except by changing the
>rules."

"unsolvable" is always relative to a known context of proving power.
A common standard is that provided by Turing machines.  One of the
basic AI debates is whether this standard suffices for the human mind.
A common argument by pro-AI-ers is that this standard suffices for the
relevant physics of the mind.  I dissent on this on two points.

First: I think quantum mechanics is relevant, with experimental backup
even.  H. Froehlich has a dozen or so papers on the existence of Bose-
Einstein condensation at high temperature in biological systems of var-
ious phonons or other more bizarre modes of excitation, using appropriate
energy pumps to maintain this quantum state (the kind of state that makes
superconductors and superfluids what they are via low temperatures).

A skim through SCI over the past decade reveals hundreds of citations.
In one paper, HF "Evidence for Coherent Excitation in Biological Systems"
_International Journal of Quantum Chemistry_, vol 23, pp1589-1595 (1983)
Froehlich mentions cyclic modes of enzymatic reactions as a model for
brain waves.  As a far out idea, I'm reminded of the BBS paper some
years back, "How brains make chaos to make sense of the world".

Second: I don't know of any proof that QM is always Turing computable. 
As an example, Pour-El and Richards prove, among other things, that for
a given effective Hilbert space, and a given sequence of complex numbers,
and a co-r.e. subset of the integers, there is a self-adjoint operator
such that its spectrum is the closure of the sequence, but its eigen-
values are exactly indexed by the given co-r.e. set.

As a QM observable, this would be a pretty hopeless beast to analyze
with a Turing machine.  As to whether this is ridiculously farfetched
or not, I have absolutely no idea.  Is it just the Schroedinger's
logician friend's cat?  Or what?

>I still don't see the need for anything beyond Schroedinger's equation
>to solve the problem.

Exactly.  But solving it is more than plugging it into your workstation,
no matter what its power.

That's the point of metamathematics: beating your head against what it
has proven is the same as trying to trisect the angle with ruler and
compass.  You can trisect angles--by changing the rules.  And similarly,
in the above scenario, you can solve the unsolvable--by changing the
rules.  (As in, the Cray Z-MP does infinite loops in only 4 seconds.)
-- 
-Matthew P Wiener (weemba@libra.wistar.upenn.edu)


