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Article 5571 of comp.ai.philosophy:
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>From: ramsay@unixg.ubc.ca (Keith Ramsay)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May12.051333.13868@unixg.ubc.ca>
Date: 12 May 92 05:13:33 GMT
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minsky@media.mit.edu (Marvin Minsky) wrote:
| The math seems generally OK, but the stuff on universal Turing 
 machines seems
|amateurish.  He either did not know, or neglected to point out that
|there are known to be very small Universal Turing Machines (e.g, 4
|symbols, 7 states).  

In article... ramsay@unixg.ubc.ca, I wrote:
|Is there some special significance to this fact (so that one would
|make a special point of including it)?

minsky@media.mit.edu (Marvin Minsky) writes:
|Yes indeed, because Penrose book is permeated by insinuations to make
|the naive reader feel that people are not machines -- and every single
|one of these insinuations is based on some sort of wrongness.  I am
|angry because *anyone* could collect a lot of defective arguments and
|then insinuate that "where there's smoke, there's fire".

I agree that it is a somewhat unfortunate book, but I disagree about
the particular case of the "largeness" of the UTM example. I don't
think it's meant to insinuate anything.

|Specifically, Penrose presents a several hundredd-digit Godel-like
|number for his universal Turing machine and then says:
|  "This number no doubt seems alarmingly large!  Indeed it *is*
|   alarmingly large but I have not been able to see how it could be
|   made significantly smaller. ... One is inevitably led to a number
|   of this size for the coding of an actual universal Turing machine."

I had another look at _The_Emperor's_New_Mind_ today, and I note that
he has a footnote at this point, reading in part:

   "Very concise-looking universal Turing machines have indeed been
   described in the literature, but the conciseness is deceptive, for
   they depend upon exceedingly complicated codings for the
   descriptions of Turing machines generally."

|Well, most of Penrose proofs have this form.  The special significance
|is that he has made you think that UTMs are so complex or something
|that a brain must be something else.  If he did not have some such
|purpose, then why did Penrose make such a special point of inculding
|it?

I don't think that was Penrose's intention.

Rather than merely assert that such a UTM exists, he decided to give a
specific example, perhaps so that people would see it as something
concrete, something one could get one's hands on. (He invites the
ambitious reader owning a home computer to examine the binary, given
in the footnote, to see that it is indeed a UTM!)

It's a bit like when people give the decimal notation for a record
size prime, or give lots of digits of pi. Note how in
_Godel_Escher_Bach_, Hofstadter also remarks how huge the
Godel-numberings of things become. Why? It's not essential
information. People seem to find it entertaining, and often seem to
like to see concrete examples.

Minsky:
|  So far as I can see, Penrose's discussion about Godel's theorem
|depends on making peculiar assumptions about (1) that humans have
|magical abilities to recognize mathematical truths

I:
|I think it is a little more subtle than that. Penrose (incorrectly,
|IMO) concludes the non-computable abilities from our ability to apply
|Godel's reasoning to arbitrary formal systems. It is more a mistake
|than an untoward assumption.

On re-reading some of Penrose's discussion, I find it vaguer than I
had remembered it being, alas.

Penrose reasons thus: (1) *we* can infer [sic! We hope not!] the Godel
sentence of any formal system, describing a machine, (2) the machine
itself can't, therefore (3) we can't be equivalent to that machine,
and (4) our process of inferring is not mechanical. This is why I
think (4) is better described as an improperly supported *conclusion*
than as an assumption. Calling it an assumption makes it sound more
like it comes out of thin air, which gives a misleading picture of
Penrose's thought processes.

Minsky:
[...]
|Are you saying that "a mistake" is better or worse than "an untoward
|assumption"?

More dangerous, because more credible, and also more liable to confuse
and distract from more serious issues. I think it is a less "stupid"
mistake on Penrose's part, however, although sloppy, than if it were
essentially a matter of his "assuming" from out of nowhere that people
have non-computational powers of recognizing mathematical truths.

Even Lucas was slightly more careful; Lucas recognizes that only for
consistent systems can the reasoning be applied, but then argues that
we *are* consistent, and so should be able to reason so.

[More criticism of Penrose's arguments]
|Well, I've been meaning to say this since college days, but I never
|got quite stirred up enough.  Thanks!

You're welcome! I plan to say more about the argument itself later.
-- 
Keith Ramsay
ramsay@unixg.ubc.ca


