Newsgroups: comp.ai.philosophy,sci.logic
Path: cantaloupe.srv.cs.cmu.edu!rochester!udel!news.mathworks.com!news.ultranet.com!news.sprintlink.net!EU.net!sun4nl!cwi.nl!olaf
From: olaf@cwi.nl (Olaf Weber)
Subject: Re: Penrose and human mathematical capabilities
In-Reply-To: tim@maths.tcd.ie's message of 13 Jul 1995 23:39:59 +0100
Message-ID: <DBp5E5.Hzp@cwi.nl>
Sender: news@cwi.nl (The Daily Dross)
Nntp-Posting-Host: havik.cwi.nl
Organization: CWI, Amsterdam
References: <3t6tcv$nca@netnews.upenn.edu> <3tkpqr$88l@bell.maths.tcd.ie>
	<jqbDBG20y.E4p@netcom.com> <3torlc$8ho@bell.maths.tcd.ie>
	<jqbDBI3sH.J50@netcom.com> <3tra4b$em9@netnews.upenn.edu>
	<3tsi61$t7g@nnrp.ucs.ubc.ca> <3ttskd$fo7@netnews.upenn.edu>
	<3tue8v$360@nnrp.ucs.ubc.ca> <3tvege$gm@bell.maths.tcd.ie>
	<DBLK7w.Fty@cwi.nl> <3u212p$5t@bell.maths.tcd.ie> <DBnFr8.CMv@cwi.nl>
	<3u47bv$cnn@bell.maths.tcd.ie>
Date: Fri, 14 Jul 1995 08:10:50 GMT
Lines: 54
Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:30147 sci.logic:12367

In article <3u47bv$cnn@bell.maths.tcd.ie>, tim@maths.tcd.ie (Timothy Murphy) writes:
> olaf@cwi.nl (Olaf Weber) writes:

>> In SotM (and TENM), Penrose spends a lot of time showing that there
>> is an isomorphism between formal systems and algorithms.

> There isn't an "isomorphism" between formal systems and algorithms.

Let's follow Matthew Wiener in amending the word to "correspondence".
It is still good enough for the present purpose.  Read Matthew's reply
to your post, or TENM or SotM or a text treating automaton theory for
the details.

>> [A] demonstration [that human beings are (in some sense) formal
>> systems] could go by showing that humans are (in some sense)
>> computers running an algorithm.

> Before you even reached that stage, you would have to agree on some
> (enumerable) encoding of whatever you took to define the configuration
> of a human being.  It's not at all obvious that such a coding is
> possible.

It isn't all that obvious that such an encoding is impossible either.
And you could get what is needed by providing an adequate algorithmic
description of a neuron.  Penrose therefore argues that neurons are
non-algorithmic "devices".

> In standard physics many variables, eg time, are supposed to take
> real values, and the real numbers are not enumerable.  So on the
> face of it the configuration-space of a human is too large to apply
> Turing theory to.

Analogue computers are still computers, and also vulnerable to Gdel's
theorem.  And in practice any analogue computer has only a limited
precision, and can therefore be adequately simulated on a digital
computer.

The human brain would, if it were a computer, be massively parallel,
asynchronous, and use a mixture of digital and analogue techniques.
This makes it rather difficult to talk about the computational state
of the brain.  But it would still be possible to model it adequately
on a von Neumann machine, for which computational states are well
defined.

> It seems to me that notions of computability only apply to a fairly
> restricted area of science and mathematics.

The difference between "real" computation (i.e., computation involving
the real numbers, rather than some approximation of them) and digital
computation seems to lie in the region of complexity: algorithms that
are intractable for TMs need not be intractable for computers that use
real numbers as single units.  The principled limits are the same.

-- Olaf Weber
