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From: pindor@gpu.utcc.utoronto.ca (Andrzej Pindor)
Subject: Re: rereRe: The end of god
Message-ID: <Cxzo7E.91v@gpu.utcc.utoronto.ca>
Organization: UTCC Public Access
References: <36vt2m$g6m@scapa.cs.ualberta.ca> <37p1h3$dbo@scapa.cs.ualberta.ca> <Cxu1yE.2vL@gpu.utcc.utoronto.ca> <383kau$5q2@scapa.cs.ualberta.ca>
Date: Thu, 20 Oct 1994 20:42:49 GMT
Lines: 62

In article <383kau$5q2@scapa.cs.ualberta.ca>,
Kevin Wiebe <kevin@sawnlk.cs.ualberta.ca> wrote:
>Andrzej Pindor (pindor@gpu.utcc.utoronto.ca) wrote:
>: In article <37p1h3$dbo@scapa.cs.ualberta.ca>,
>: Kevin Wiebe <kevin@swanlake.cs.ualberta.ca> wrote:
>: >..........
>: >Example: Mathematics can prove it is incomplete, itself.  We know of
>: >a polynomial 'D' which is of degree eight, with integer coefficients, 
>: >and that has eighty variables that has no solutions; but it is impossible
>: >to prove this within mathematics!  Namely, D(x_1, x_2, ..., x_80) = 0
>: >has no solutions in the natural numbers (we can show this for sure), 
>:                                              ^^^^^^^^^^^^^^^^^^^^^^     
>: >but we can't prove it mathematically - mathematics is incomplete.
>: >(I will not type 'D' here, because at least one of the coefficients
>: >is thousands of digits long.)
>: >
>: Forgive a naive question, but how "showing this for sure" is different from
>: "proving within mathematics"?
>
>: >Harrington, Paris, and Kirby recently discovered a true statement "Ra"
>: >about natural numbers that is not provable from mathematics.  
>: >
>: Did they discover that this statement is true without using mathematics?
>: What did they use?
>: Just curious.
>
>: Andrzej
>
>Ah, I was wondering when someone would get to this point.  This is
>where my explanation skills run dry.  I cannot clearly explain how
>we "see" these truths we cannot prove within the system, but maybe
>someone else can explain this clearly????
>
>Basically, we can satisfy ourselves that the statements are true
>by examening them outside the system - but we cannot prove them
>within the system itself.  Maybe an example will help - maybe not.
>
>EXAMPLE:
>========
.............
(deleted for brevity)
>
>In any case, there is an example of how we can "know" something is true
>without having to remain within the system which cannot prove its truth.
>-Kevin-

Your example just illustrates the Goedel theorem.  The point I was trying to 
make was that to know something 'for sure' we also use mathematics, even
if applied to a system external to the one in which this something is true.
Short of divine inspiration, what we hold to be true in science is arrived at
by logical reasoning at some level. We may propose various conjectures and
even have a deep, unfaltering belief that such a conjecture is true, it only
becomes a scientific truth if proven using logic. Penrose seems to suggest
that there are some scientific (mathematical) truths which logic cannot prove.
I have yet to hear an example. Yours does not cut it.

Andrzej
-- 
Andrzej Pindor                        The foolish reject what they see and 
University of Toronto                 not what they think; the wise reject
Instructional and Research Computing  what they think and not what they see.
pindor@gpu.utcc.utoronto.ca                           Huang Po
