From: axs@cs.bham.ac.uk (Aaron Sloman)
Newsgroups: comp.ai.philosophy
Subject: Re: Human intelligence vs. Machine intelligence
Date: 15 Oct 92 22:57:18 GMT
Organization: School of Computer Science, University of Birmingham, UK

daryl@oracorp.com (Daryl McCullough) writes:

> Date: 14 Oct 92 12:23:02 GMT
> Organization: ORA Corporation
> ...
>
> To summarize again the facts of this matter, Penrose' argument goes as
> follows:
>
> 1. For every consistent formal system T powerful enough to formalize
> arithmetic, there is a sentence G of arithmetic such that (a) G is true,
> and (b) G is not provable by T.

I know this is the standard interpretation of Go"del's argument. But
it seems to me that there's a two-fold problem that both Penrose and
his critics haven't noticed.

(i) When you look very closely at the Go"del sentence G, it is
actually a sentence in which a complex recursively defined predicate
is applied to a very large number. Most people take G as *saying*
that the sentence G itself is not derivable within T. This is the
reason why they conclude that (if T is consistent) then G is true,
because Go"del proved that G (and its negation) are not derivable in
T (if T is consistent). If that's what he proved and that's what G
said then yes, if T is consistent then G is true.

But there's a subtle slide here because a sentence about properties
of a number is NOT a sentence about properties of the formal system
T. Of course Go"del tried using the Go"del-numbering scheme to set
things up so that his sentence G (about the number) would be true if
and only if the corresponding sentence about derivations in T were
true. But it is not obvious to me that this is going to be so in all
models of T. Especially in the light of (ii), below.

(ii) If, as Go"del proved, G and not-G are both not derivable in T
(if T is consistent), then if T is consistent then so are both
    T + G,
and
    T+ not-G.
That means both will have models. That means that in some models
of T G will be true and in some models G will be false, if T is
consistent.

This seems to me to undermine anyone's claim to be able to *know*
that G is true, whether it is Penrose (who says he does and
computing machines can't) or his critics (some of whom say they do
and computing machines can also).

Of course some people could argue against this that G is true in the
model they are thinking of (the "intended" model). But then they
would have to specify *how* they are able to think of that
particular model. (Most people find it intuitively "obvious" that
there is one "right" model for arithmetic. But that's exactly what's
debatable.) Go"del shows, in effect that they can't identify their
model by having a particular (consistent) formal system in mind: no
(consistent) formal system of the required complexity will suffice
to determine the model (because it is consistent with both G and
not-G).

If they claim they can specify the "right" model by some other means
than a formal system then either they are deluding themselves, or
this other means must support the position of Penrose, ie. that
there's something other than a standard sort of formal system
underlying human arithmetical abilities.

My guess is that they are all deluding themselves!

This is (part of) the argument I offered against both Penrose and
some of his critics in the review that Matt Ginsberg recently
claimed he found so embarrassingly flawed. However I have to say
(and indicated in the review) that I find the argument very
complicated and am not sure whether it has technical faults. (I've
shown it to various logicians who apparently did not think so, but
that proves nothing.)

> 2. From 1., if we know that a system T is consistent, then we know
> that the corresponding sentence G is true.

If my argument is correct, then we wouldn't know this at all. The
illusion that we know this is based on the slide from the sentence G
(about a number) to the metamathematical statement about T. If (i)
above is correct, then this slide is unjustified. (Presumably
because the equivalence that Go"del tried to set up would hold only
in *some* models of T!).

> 3. Since we know that G is true, and T cannot prove G, then we know
> something that T can't prove.
>
> 4. Therefore T cannot be a formalization of our reasoning.
>
> The conclusion of this argument is that no system known by us to be
> consistent can completely capture human reasoning. It does not say
> that no system can completely capture human reasoning, and it does not
> say that no consistent system can completely capture human reasoning.
> It says that no system *known by us to be consistent* can capture all
> of human reasoning.

If the points above are correct then it doesn't even say this much!

> ...In other words, proofs (or even good arguments
> for) consistency are extremely difficult things; it is impossible to
> demonstrate the consistency of a formal system without using an even
> more powerful formal system.

Penrose thinks that the consistency of the ways that mathematicians
think about arithmetic has been established beyond all reasonable
doubt. It's a sort of empirical argument, as I interpret him.

Even if he is correct about this, he can't have his conclusion if my
arguments are correct. So I think my attack on him is stronger than
yours!

> I have already given the example of Quine's NF set theory. To see how
> completely wrong Penrose' argument is, try to use Godel's theorem to
> show that there is some arithmetic statement that humans know to be
> true, but which NF cannot prove. You can come up with the Godel
> statement G for NF, but you don't know whether G is true unless you
> know whether NF is consistent (which nobody knows). You are stuck.
>
> There are three possible situations:
>
> 1. NF is inconsistent, and eventually we will prove it inconsistent.

I don't really know what Penrose thinks about NF, but from my
understanding of his position I think he would say that if NF turned
out to be inconsistent then that would merely show that it can't be
a formalisation of "standard" mathematical ways of thinking about
numbers, since (according to him) there's no doubt that these are
consistent. (I am being vaguer than he would be, probably.)

Incidentally, it has been suggested to me that an argument closely
related to my points (i) and (ii) can be found buried in
Hofstadter's Godel Escher Bach, though he interleaves so many
threads in that (marvellous) book that it is hard to check such
claims.

I do not claim that my arguments are original. On the contrary, I
have vague recollections of similar lines of argument many years
ago when I was a logic student in Oxford, but I don't have any
references. Moreover, I think my arguments may be related to the
position of the mathematical intuitionists (Brouwer, Heyting)
according to which (roughly speaking) it's not the case that our
concept of arithmetic (or any other infinite system) is fully
determinate so that every sentence in it must either be true or be
false. For some reason Penrose (who must know about these things)
never mentioned this in his book. Perhaps because it undermines his
position so strongly?

Aaron
-- 
Aaron Sloman, School of Computer Science,
The University of Birmingham, B15 2TT, England
EMAIL   A.Sloman@cs.bham.ac.uk  OR A.Sloman@bham.ac.uk
Phone: +44-(0)21-414-3711       Fax:   +44-(0)21-414-4281
Newsgroups: comp.ai.philosophy