Newsgroups: comp.ai.philosophy
From: Lupton@luptonpj.demon.co.uk (Peter Lupton)
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!news.mathworks.com!newshost.marcam.com!usc!howland.reston.ans.net!pipex!demon!luptonpj.demon.co.uk!Lupton
Subject: Penrose's Argument
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Date: Wed, 16 Nov 1994 12:54:18 +0000
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Reading the 'Shadows of the Mind' is an interesting experience.
Penrose claims to produce a reductio ad absurdum. I must 
confess to only finding one such argument -- that Penrose
has produced a reductio ad absurdum has plainly, on Penrose's
own account, been shown to be false.

Various contributers have pointed out various 'loop-holes'
that Penrose has not closed. I would like to add one more.

Suppose we accept the thrust of Penrose's argument to the
effect that 'human mathematicians are not using a knowably
sound algorithm in order to ascertain mathematical truth'
(page 76 SOTM).
[I am aware that some will rail against this initial step,
pointing out that the very idea of 'unassailable 
mathematical truth' is incoherent, but bear with me.]
Suppose we accept all this. What follows?

Well, we can say that there may be an algorithm but that
it is unknowable in the sense of 'know' required for Penrose's
                 ---------------------------------------------
use of Goedel's theorem. 
------------------------

The sense of the word 'know' required by Penrose is plainly very
strict - this sort of knowledge must be of the unassailable
sort in order for the Goedel statement derived from that algorithm
to be of the unassailable sort. The point is, that what mathematicians
actually do could only be known to us through inductive processes of
one sort or another - the detection and projection of regularities
in the behaviour of mathematicians. The algorithm we are talking about
may well be, for example, the biosphere of the earth! 

Inductive knowledge, however, could not be known to be true 
unassailably - we have known that since Hume. Hume's argument
that the inference from the seen to the unseen is (from a 
deductive point of view) irredeemably invalid is, after all,
a sound argument. 

So Penrose is not *just* asking us to accept Platonic truths in
the mathematical sphere - for which there may be some support.
No. How is also asking to accept Plantonic truths in physics -
the idea that certain inductive assertions (such as this is the
algorithm by which maths is done) could be known unassailably.

I find this a little difficult to swallow.

Cheers,
Pete Lupton

