From: Jim Balter 
Date: Fri, 22 Nov 1996 04:14:47 -0800
Organization: JQB Enterprises

Phil Roberts, Jr. wrote:
> 
> On Wed, 13 Nov 1996 22:03:17 GMT
> Jim Balter wrote:
> >
> >In article <328A2BEC.70E1@ix.netcom.com>,
> >Phil Roberts, Jr.  wrote:
> 
> >>You're right.  Lucas is _trivially_ wrong, in that he referred to his
> >>attempt to extend Godel to the material realm as a _proof_ of the falsity
> >>of mechanism (last paragraph).  What he offered was merely an argument,
> >>but a pretty damn good one, if I'm not mistaken:
> >
> >You are very mistaken.
> >
> >>      Godel's theorem states that in any consistent system
> >>      which is strong enough to produce simple arithmetic
> >>      there are formulae which cannot be proved-in-the-system,
> >>      but which we [standing outside the system] can _see_ to
> >>      be true.
> >
> 
> Its important to understand that this is not Lucas's idiosyncratic view of
> Godel.  This _is_ what Godel's theorem amounts to when translated into
> simple English.

Godel's theorem does not refer to "we", thus this must be false.

I, for one, have never seen to be true the Godel sentence of any system;
in fact I've never seen such a sentence.  Have you?  Can you state one?

If I did see one, I certainly wouldn't just "see" that it is true,
but would have to have that proven to me.  Of course, the proof
could not be given in the system for which the sentence is the
Godel sentence.

>         Godel's Theorem says that in any consistent axiom system rich
>         enough to generate the arithmetic of the natural numbers, there
>         are statements we cannot prove in the system, but which can be
>         _seen_ by other means to be true (Brainstorms, p. 257).

Dennett does not emphasize "seen", and it is just plain dishonest
of you to add that emphasis.  Dennett doesn't say how this seeing
is to be done, or what it means.

>         ...what Godel proved, _beyond any doubt_, is that when it comes
>         to axiomatizing simple arithmetic, there are truths that "we
>         can see" (Godel) to be true but that can _never_ be formally
>         proved to be true (Darwin's Dangerous Idea, p. 429).

And then he immediately says "Actually this claim must be carefully
hedged: ...".  This hedge includes consistency and, more importantly,
that this "seeing" is a matter of changing systems and proving
(quite formally, thank you) the truth not provable in the previous
system.

He later goes on to say of the idea that we can "see" truths
that we can't prove, "But although this seems to be what Godel
himself thought, and it certainly expresses the general popular
understanding of what Godel's Theorem shows, it is much harder
to *demonstrate* than first appears".

IOW, Godel's Theorem does not itself prove that we can "see" these
truths; that's a inference drawn by Godel and others, but is *not
proven*.  (In fact, Godel backed off; see "1951" below).

So, what you say is important to understand above is in fact false and
is contradicted by your own sources.

> Its also important to understand that Godel's theorem (and the above simple
> English statement of that theorem) is all but universally regarded as
> valid or "true" by mathematicians:

Yes for Godel's theorem, no for the parenthesized claim. 
 
>         The technical details of Godel's proof itself need not concern us;
>         *no mathematician doubts its soundness* (ibid p. 429).

Right, but there are plenty of doubts of what this says about us
and what we can see.

> >First, for any such system, there is another consistent system in which the
> >Godel sentence of the first system can be proven.
> 
> Which in turn has its Godel sentence which can not be proven and on and on.

Yes; and the same might be true for us.  We can't see the truth of
our own Godel sentence; we don't even have the faintest idea what it
is, and in fact if we aren't governed by Godel then we don't have one.

> > The "we" here is a red herring.
> 
> The point is that we can "see" truths that neither we nor our computers can
> prove.  You need a "we" to do the seeing, at least for the time being, since
> "we" are the only systems we know of at the moment that can accomplish this
> feat.

This is what is to be proven; Lucas didn't do it, nor did Godel.
While the details of Godel's theorem needn't concern Dennett,
it does need to concern you if you are going to claim that it
show something about what we can "see".  I suggest that you
read it.

> >In addition, inconsistent systems of the same strength can also
> >"prove" that Godel sentence.
> 
> Formal systems which prove theorems employing inconsistent logic?  That's
> a new one on me.  I'm all ears.

So what if it is new to you?  That doesn't speak in your favor.
Any proof system with inconsistent premises can prove all sorts of
things.  In practice, most of our machines do not implement consistent
systems; they are heuristic, thus they can make errors.  Just like us.

> >Second, although we can "see" that the Godel sentence of any such system must
> >be true, we cannot necessarily see the Godel sentence *as* true.  That is, if
> >presented with the Godel sentence, there is no reason to think that we will be
> >able to see that it is true, unless we can show that it *is* the Godel
> >sentence for that system.
> 
> In other words, the truth we can "see" is not some universal truth, but one
> which, in this case, is context dependent and limited in its scope.

No, I didn't say anything like that.

> I don't
> have a problem with that.  But it still leaves the argument untouched.  

What argument?

> We
> can just "see" a truth (albeit context dependent and limited in scope) that
> neither we nor our machines can prove.

That's just an assertion.  I can just "see" that this does not
follow from Godel's Theorem (it certainly wasn't proven by Godel;
read the theorem).

> >And, just as we can "see" that Godel's theorem is
> >true (after all, it is a theorem),
> 
> I'm not so sure we can.

Huh?  It's a *theorem*, so we can see that it is true.

> But its irrelevant.  The point is that Godel
> has _proven_ his theorem, and what it says about the nature of cognition
> appears to be fairly startling, that we can "see" truths that can't be
> proven.

As Dennett pointed out, it doesn't say this at all.  This is just
an inference a lot of people make, an unproven inference.

And it's relevant to the argument I was presenting, which you
show no signs of understanding.

> >there are, contrary to popular belief,
> >formal consistent systems in which Godel's theorem can be derived (after all,
> >it is a theorem).
> >
> 
> Correct but irrelevant.

Of course it is relevant.  You just don't understand the relevance.

> >It may be that "we" could be modeled as a sufficiently strong consistent
> >system, in which case, per Godel, we could not see that our own Godel
> >statement is true, even though we can see that our Godel sentence must be
> >true.
> 
> I think you mean we could not prove our own Godel sentence, even though we
> can see that it must be true.

No, I mean what I said, which you did not understand.  We can see
that our Godel sentence, whatever it is, if we have one, must be true;
we can *prove* that; Godel did.  But we can't see or prove that our
Godel sentence is true, because we don't even know what it is or
whether we have one.  If presented with our own Godel sentence, if we
have one, we would not see that it is true.  We would not recognize it
as our Godel sentence.  This follows from Godel's proof.

>  But that's just a repetition of Godel itself,
> since it merely refers to truths which can not be proven, by humans _or_
> machines.

Godel's Theorem says nothing about what we can see.  It is a formal
result that shows that, given a specific consistent system of a
certain sort, there is a sentence expressible in that system
that is consistent with the premises of the system but which
cannot be proven within that system.  That's it; the rest of this
stuff about what *we* can see is not to be found in Godel's theorem,
but only in his metaphysical musings.

> >This is not a contradiction; rather it implies that we would not be
> >able to recognize or derive our own Godel sentence.
> 
> We might well be able to _recognize_ it, but we would not be able to
> derive (prove) it, since that's precisely what Godel's theorem says.

How?  How could we recognize it?  Have you even the faintest idea
of what your Godel sentence should look like?  Hmmm?  Now, given a
system able to prove Godel's theorem, we know that that system
cannot derive its own Godel's sentence, because if it could it would
follow from Godel's theorem that that sentence was true, but that's
a contradiction, because Godel's theorem says that can't be made to
follow.  This is a simple ad reductio.

> >This is not too terribly
> >subtle, but it is too much for Lucas and many of his readers.
> 
> If by that you mean those who agree with Lucas, they include both Godel and
> Penrose, a couple of pretty dim bulbs in your book, eh?

No, I meant what I said: "many of his readers".  I meant to include
you.  Godel didn't use Lucas' argument, and Penrose had to come up
with a much more sophisticated argument in _Shadows of the Mind_
after being thrashed for _The Emperor's New Mind_.  Such "dim bulbs"
as Turing, Putnam, and Feferman reject Lucas, and I could name
many many others.  Penrose is a good physicist, but his argument
comes down to believing that there can be unprovable yet unassailable
truths ("informal and guaranteed accurate", as he puts it); I don't
have to accept his metaphysical commitments just because he's another
bright guy, and I consider myself in very good company in this regard.
I recommend

    http://psyche.cs.monash.edu.au/psyche-index-v2_1.html

for rebuttals to Penrose's arguments and his response.  One thing
notable here is how much more complex this argument is than just
say "Lucas showed it" or "Godel proved it"; that's because they didn't.

>         Godel himself thought that the implication of his theorem was
>         that human beings -- at least the mathematicians among us --
>         cannot, then, be just machines, because they can do things no
>         machine can do (ibid., p 430).

But this is merely what he thought, not what he proved, and he had
second thoughts.  Solomon Feferman, the mathematician whose work Penrose
depended upon, offers his own rebuttal of Penrose on the page mentioned
above, and here is what he has to say there about Godel's beliefs:

  Penrose reports in section 3.1 on what Gödel took the significance of
  his incompleteness theorems to be, via a quotation which had
  circulated some time back from Gödel's unpublished Gibbs lecture of
  1951. That piece is now available in full as *1951 in Gödel (1995),
  with an illuminating introductory note by George Boolos. More cautious
  than Penrose, Gödel there comes to the conclusion that "either...the
  human mind (even within the realm of pure mathematics) infinitely
  surpasses the powers of any finite machine, or else there exist
  absolutely unsolvable diophantine problems." 

Do you understand the import of that "or else"?  Probably not.
You just like the conclusion that minds are more powerful than
machines, and quote bits and pieces of arguments that fit your
metaphysical commitments.  In this you are like Penrose.

> >>      Godel's theorem must apply to cybernetical machines,
> >>      because it is of the essence of being a machine, that
> >>      it should be a concrete instantiation of a formal
> >>      system.  It follows that given any machine which is
> >>      consistent and capable of doing simple arithmetic,
> >
> >This is confused.  The system *implemented* by a Turing machine need not
> >be consistent, nor strong enough for Godel's purposes.
> 
> If consistent systems strong enough for Godel's purposes can't prove the
> Godel sentence, then all the worse for inconsistent systems or systems
> insufficient for simple arithmetic.  Unless, of course, you can show
> us how to prove theorems using inconsistent logic, which you seem to
> think possible (above).

You are completely out of your depth; (P and not P) -> Q, for any Q.
This is *really* basic.  It is the *consistency* of the systems
Godel considers that results in the limitation.  Godel's result is
that these systems cannot be both consistent and complete; they
can be consistent and incomplete, *or* they can be inconsistent and
complete.  This is really duhville; you need to get a *basic* education
in the subject.

> >>      there is a formula which it is incapable of producing
> >>      as being true -- but which we can _see_ to be true.
> >
> >Again, a red herring; Godel sentences are *per system*.  Each system fails to
> >be able to derive its *own* Godel sentence.  Even if we can see the Godel
> >sentence of a system to be true, that says nothing about our *own* GS, if we
> >have one.
> 
> Yes it does.  It says we can "see" it even though we can't prove it.

You are out of your depth.  You don't even understand what "it" is
here.

> >In addition, we *cannot* see the truth of the vast majority systems'
> >GS's; they are way too complicated.  We *can* see the truth of Godel's
> >theorem, but *that's a different matter*.
> >
> >>      It follows that no machine can be a complete or adequate
> >>      model of the mind, that minds are essentially different
> >>      from machines.
> >
> >Totally wrong.  There are consistent systems that can derive Godel's theorem.
> >Nonetheless, they cannot derive their own Godel sentences.
> >Lucas does not show that we cannot be such a system.
> >
> 
> Its not Lucas, but Godel that concerns us here.  And he has actually _proven_
> that, although we can prove Godel's theorem, neither we nor our machines
> can _prove_ our Godel sentences.

You are out of your depth.  If Lucas is right,we don't *have*
Godel sentences.


> So why would Lucas want to show that we
> ourselves can do something (prove our G sentence) that the theorem from
> which he derives his argument says we cannot do?

Because his point is that we are not the sorts of systems that
Godel's theorem applies to.  Duh.  You are *way* out of your depth.
 
> >Now, those are a host of Lucas' more subtle failures.  But the easy, trivial,
> >refutation, as I and Turing before me stated, is that both machines and minds
> >can implement inconsistent systems, to which Godel does not apply.
> >
> 
> True.  Nor can they prove theorems.  Unless, of course, you plan to show us
> how.

Premise 1: p
Premise 2: not p
Suppose not Fermat's last theorem
	p (premise 1)
	not p (premise 2)
	contradiction
Conclusion: Fermat's last theorem (reductio ad adsurdum)

This is freshman logic.  You are way way out of your depth.

> >The bottom line is, Lucas gets almost everything wrong, and it is a pathetic
> >waste of time to build anything upon such ill-informed error.
> >--
> >
> 
> Why then do you think two of the century's most eminent mathematicians
> have, indeed, wasted so much time on it?

Godel had nothing to say about Lucas (who came after him), and
Penrose's expertise is physics; his mathematics is not particularly
outstanding.  There are many better mathematicians (as Penrose would
readily agree, and probably says as much somewhere in his books),
including Penrose's mathematical mentor Solomon Feferman mentioned
above, who have a rather different opinion.  Don't judge someone's
mathematical competence by how many of their books you can find in the
popular science section of the local bookstore.  And Turing, whose work
is in many ways an improvement upon, and certainly more significant
than, Godel's work, took the opposite position.  But you just like to
pick and choose what fits your preconceptions.  I wrote of Lucas's
errors; vague appeals to "eminent mathematicians" are no way to make an
error not an error.

> What truth is it that you can
> just "see" that seems to have evaded both Godel and Penrose, and me,
> for that matter (at least I'm in pretty good company)?

I gave arguments.  It's not my fault that you can't follow them
and thus must depend upon throwing these names around.  If you
like names, read the ones on the articles mentioned in the above
reference.

> I think its important to keep in mind the fact that there are actually
> three issues at play here:
> 
> 1. The validity of Godel's theorem (including what it says, I suppose).

The validity of Godel's theorem is not in doubt.
What it says, though, is essential to any argument built
upon it.

> 2. The validity of Lucas' argument that it applies to material systems,
>    and therefore that minds are different from theorem proving machines.

Minds are different from theorem proving machines, but not for
the reasons Lucas gives.  One point is that most formal systems,
most machines, practically the whole subject of AI, has nothing to
do with theorem proving.  Even if Penrose/Lucas are right, it would
only mean that certain activities of certain mathematicians could
not be performed by machines; all other human behavior is left untouched
by these arguments.

> 3. The testability of Lucas' argument.

It isn't testable, any more than the theorem that there are
an infinite number of primes is testable.  The claim that there
are things that humans can do that no machine can do could only
be "tested" by building all possible machines.
 
> Its actually 3 which is where Dennett thinks Lucas and Penrose are most
> vulnerable (ibid, Chapt 15),

You have misunderstood Dennett. 

> and which is why I have assumed my own
> theory and the _empirical_ _evidence_ I have presented should be of
> some small interest to thoseworking in strong A.I.

The Lucas argument is not empirical; it's validity or soundness
does not and cannot depend upon empirical evidence, any more than
the argument for an infinity of primes.  You are *so* far out of your
depth ....

--