Newsgroups: comp.ai.philosophy,sci.logic
Path: cantaloupe.srv.cs.cmu.edu!bb3.andrew.cmu.edu!nntp.sei.cmu.edu!news.psc.edu!hudson.lm.com!news.math.psu.edu!news.cac.psu.edu!newsserver.jvnc.net!newsserver2.jvnc.net!howland.reston.ans.net!news.nic.surfnet.nl!sun4nl!cwi.nl!olaf
From: olaf@cwi.nl (Olaf Weber)
Subject: Re: Penrose and human mathematical capabilities
In-Reply-To: weemba@sagi.wistar.upenn.edu's message of 14 Jul 1995 14:25:37
	GMT
Message-ID: <DBpzJI.3nu@cwi.nl>
Followup-To: comp.ai.philosophy,sci.logic
Sender: news@cwi.nl (The Daily Dross)
Nntp-Posting-Host: zeus.cwi.nl
Organization: CWI, Amsterdam
References: <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> <DBp5E5.Hzp@cwi.nl>
	<3u5up1$2ji@netnews.upenn.edu>
Date: Fri, 14 Jul 1995 19:02:02 GMT
Lines: 90
Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:30220 sci.logic:12432

In article <3u5up1$2ji@netnews.upenn.edu>, weemba@sagi.wistar.upenn.edu (Matthew P Wiener) writes:
> In article <DBp5E5.Hzp@cwi.nl>, olaf@cwi (Olaf Weber) writes:

> I'd say you're being simplistic.  Neurons are susceptible to
> chemistry, a more recently discovered volume channel.  And
> immunological status is known to effect our minds.

As do hormones.  But chemical influence ultimately resolves into the
presence or absence of single molecules on recoptors.

> Whatever is going on overall is from from being plausibly digital.

It doesn't need to be digital, as long as it could be adequately
modeled on a digital computer.

>> Analogue computers are still computers, and also vulnerable to
>> Gdel's theorem.

> Uh, no.  Goedel's theorem is not about "computers", per se.

I know, Goedel's theorem is about formal systems, and its applicability
to computers depends on a mapping of formal systems to computers.  But
there are no indications that I know of that analogue computers won't
have the same kinds of restraints that hold for digital computers.  If
you do, please give references, as I'd like to look up the papers.

>> And in practice any analogue computer has only a limited
>> precision, and can therefore be adequately simulated on a digital
>> computer.

> Again, no.  Analog possibilities that violate Church's thesis are
> known.

Examples and/or references?

As the matter stands, I expect that for a definition of analog
computation, people will find a similar thesis to Church's thesis, and
that the formal systems by which analog computers are modeled will be
as vulnerable to Goedel-type arguments as those modeling digital
computers.

> Whether we could build such devices is another question.

If we cannot build them, is there any reason that such devices can
exist at all?  If not, then they are irrelevant to any argument
against AI.

>> 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.

> This does not follow whatsoever.  There is no a priori reason to
> believe that physics is computational, so appealing to such a reason
> is nonsense.

Note the hedging: "if [the human brain] were a computer".  That's why
the rest follows.

By the way, there are a priory reasons that the human brain was shaped
by an alogithmic process, evolution.  Exactly how that process managed
to construct a non-algorithmic enitity is something to be explained by
people who hold that the brain is such an entity.

>> 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.

> That is not clear at all.  The field is still in its infancy, and even
> what the correct definitions should be--let alone which models what we
> could build--is debatable.

I'm merely reporting what an (incomplete) reading of the literature
indicates.  I haven't seen any argument that analogue computers could
do things that turing machines cannot, but perhaps you can refer me to
a paper where this is demonstrated.

One reason to doubt that there will be much difference between the
capabilities of analogue and digital systems is that _we_ are quite
incapable of representing irrational numbers except by symbolic name
or recipy, and yet manage to reason rigorously about them.  So an
argument that you need to have "real" irrational numbers because you
cannot reason about them "by proxy" seems to be fallacious.

-- Olaf Weber
